I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings
Let R be a ring, let I be an ideal of R, and let n be a fixed positive integer. We define and study I−n-injective modules and I−n-flat modules. Moreover, we define and study left I−n-coherent rings, left I−n-semihereditary rings, and I-regular rings. By using the concepts of I−n-injectivity and I−n-...
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irk-123456789-1660842020-02-19T01:26:54Z I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings Zhanmin, Zhu Статті Let R be a ring, let I be an ideal of R, and let n be a fixed positive integer. We define and study I−n-injective modules and I−n-flat modules. Moreover, we define and study left I−n-coherent rings, left I−n-semihereditary rings, and I-regular rings. By using the concepts of I−n-injectivity and I−n-flatness of modules, we also present some characterizations of the left I−n-coherent rings, left I−n-semihereditary rings, and I-regular rings. Нехай R — кільце, I — ідеал R, а n — фіксоване додатне цілє число. Ми визначаємо та вивчаємо I−n-ін'єктивні модулі та I−n-плоскі модулі. Крім того, визначаємо та вивчаємо ліві I−n-когерентні кільця, ліві I−n-напівспадкові кільця та I-регулярні кільця. За допомогою концепцій I−n-ін'єктивності та I−n-пологості модулів також наводимо деякі характеристики лівих I−n-когерентних кілець, лівих I−n-напівспадкових кілець та I-регулярних кілець. 2014 Article I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings / Zhu Zhanmin // Український математичний журнал. — 2014. — Т. 66, № 6. — С. 767–786. — Бібліогр.: 2 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166084 512.5 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Zhanmin, Zhu I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings Український математичний журнал |
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Let R be a ring, let I be an ideal of R, and let n be a fixed positive integer. We define and study I−n-injective modules and I−n-flat modules. Moreover, we define and study left I−n-coherent rings, left I−n-semihereditary rings, and I-regular rings. By using the concepts of I−n-injectivity and I−n-flatness of modules, we also present some characterizations of the left I−n-coherent rings, left I−n-semihereditary rings, and I-regular rings. |
| format |
Article |
| author |
Zhanmin, Zhu |
| author_facet |
Zhanmin, Zhu |
| author_sort |
Zhanmin, Zhu |
| title |
I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings |
| title_short |
I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings |
| title_full |
I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings |
| title_fullStr |
I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings |
| title_full_unstemmed |
I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings |
| title_sort |
i−n-coherent rings, i−n-semihereditary rings, and i-regular rings |
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Інститут математики НАН України |
| publishDate |
2014 |
| topic_facet |
Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166084 |
| citation_txt |
I−n-Coherent rings, I−n-semihereditary rings, and I-regular rings / Zhu Zhanmin // Український математичний журнал. — 2014. — Т. 66, № 6. — С. 767–786. — Бібліогр.: 2 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT zhanminzhu incoherentringsinsemihereditaryringsandiregularrings |
| first_indexed |
2025-07-14T20:44:08Z |
| last_indexed |
2025-07-14T20:44:08Z |
| _version_ |
1837656552645853184 |
| fulltext |
UDC 512.5
Zhu Zhanmin (Jiaxing Univ., China)
I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS
AND I-REGULAR RINGS
I-n-КОГЕРЕНТНI КIЛЬЦЯ, I-n-НАПIВСПАДКОВI КIЛЬЦЯ
ТА I-РЕГУЛЯРНI КIЛЬЦЯ
Let R be a ring, I an ideal of R and n a fixed positive integer. We define and study I-n-injective modules, I-n-flat modules.
Moreover, we define and study left I-n-coherent rings, left I-n-semihereditary rings and I-regular rings. By using the
concepts of I-n-injectivity and I-n-flatness of modules, we also present some characterizations of left I-n-coherent rings,
left I-n-semihereditary rings, and I-regular rings.
Нехай R — кiльце, I — iдеал R, а n — фiксоване додатне цiле число. Ми визначаємо та вивчаємо I-n-iн’єктивнi
модулi та I-n-плоскi модулi. Крiм того, визначаємо та вивчаємо лiвi I-n-когерентнi кiльця, лiвi I-n-напiвспадковi
кiльця та I-регулярнi кiльця. За допомогою концепцiй I-n-iн’єктивностi та I-n-пологостi модулiв також наводимо
деякi характеристики лiвих I-n-когерентних кiлець, лiвих I-n-напiвспадкових кiлець та I-регулярних кiлець.
1. Introduction. Throughout this paper, n is a positive integer, R is an associative ring with identity,
I is an ideal of R, J = J(R) is the Jacobson radical of R and all modules considered are unitary.
Recall that a ring R is called left coherent if every finitely generated left ideal of R is finitely
presented; a ring R is called left semihereditary if every finitely generated left ideal of R is projective;
a ring R is called von Neumann regular (or regular for short) if for any a ∈ R, there exists b ∈ R
such that a = aba. Left coherent rings, left semihereditary rings, von Neumann regular rings and
their generalizations have been studied by many authors. For example, a ring R is said to be left
n-coherent [1] if every n-generated left ideal of R is finitely presented; a ring R is said to be left
J-coherent [8] if every finitely generated left ideal in J(R) is finitely presented; a ring R is said to be
left n-semihereditary [24, 25] if every n-generated left ideal of R is projective; a ring R is said to be
left J-semihereditary [8] if every finitely generated left ideal of R is projective; a commutative ring
R is called an n-von Neumann regular ring [14] if every n-presented right R-module is projective.
In this article, we extend the concepts of left n-coherent rings and left J-coherent rings to left
I-n-coherent rings, extend the concepts of left n-semihereditary rings and left J-semihereditary
rings to left I-n-semihereditary rings, and extend the concept of regular rings to I-regular rings,
respectively. We call a ring R left I-n-coherent (resp., left I-n-semihereditary, I-regular) if every
finitely generated left ideal in I is finitely presented (resp., projective, a direct summand of RR). Left
I-1-coherent rings and left I-1-semihereditary rings are also called left I-P -coherent rings and left
IPP rings respectively.
To characterize left I-n-coherent rings, left I-n-semihereditary rings and I-regular rings, in Sec-
tions 2 and 3, I-n-injective modules and I-n-flat modules are introduced and studied. I-1-injective
modules and I-1-flat modules are also called I-P -injective modules and I-P -flat modules respec-
tively. In Sections 4, 5, and 6, I-n-coherent rings, I-n-semihereditary and I-regular rings are in-
vestigated respectively. It is shown that there are many similarities between I-n-coherent rings and
coherent rings, I-n-semihereditary rings and semihereditary rings, and between I-regular rings and
regular rings. For instance, we prove that R is a left I-n-coherent ring ⇔ any direct product of I-
c© ZHU ZHANMIN, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6 767
768 ZHU ZHANMIN
n-flat right R-modules is I-n-flat⇔ any direct limit of I-n-injective left R-modules is I-n-injective
⇔ every right R-module has an I-n-flat preenvelope; R is a left I-n-semihereditary ring⇔ R is left
I-n-coherent and submodules of I-n-flat right R-modules are I-n-flat⇔ every quotient module of an
I-n-injective left R-module is I-n-injective⇔ every left R-module has a monic I-n-injective cover
⇔ every right R-module has an epic I-n-flat envelope; R is an I-regular ring⇔ every left R-module
is I-P -injective⇔ every left R-module is I-P -flat⇔ R is left IPP and left I-P -injective.
For any module M, M∗ denotes HomR(M,R), and M+ denotes HomZ(M,Q/Z), where Q is
the set of rational numbers, and Z is the set of integers. In general, for a set S, we write Sn for the
set of all formal (1 × n)-matrices whose entries are elements of S, and Sn for the set of all formal
(n× 1)-matrices whose entries are elements of S. Let N be a left R-module, X ⊆ Nn and A ⊆ Rn.
Then we definite rNn(A) = {u ∈ Nn : au = 0 ∀a ∈ A}, and lRn(X) = {a ∈ Rn : ax = 0 ∀x ∈ X}.
2. I-n-injective modules. Recall that a left R-module M is called F -injective [11] if every R-
homomorphism from a finitely generated left ideal to M extends to a homomorphism of R to M, a
left R-module M is called n-injective [16] if every R-homomorphism from an n-generated left ideal
to M extends to a homomorphism of R to M, 1-injective modules are also called P -injective [16],
a ring R is called left P -injective [16] if RR is P -injective. P -injective ring and its generalizations
have been studied by many authors, for example, see [16, 17, 19, 22, 26]. A left R-module M
is called J-injective [8] if every R-homomorphism from a finitely generated left ideal in J(R) to
M extends to a homomorphism of R to M. We extends the concepts of n-injective modules and
J-injective modules as follows.
Definition 2.1. A left R-module M is called I-n-injective, if every R-homomorphism from an
n-generated left ideal in I to M extends to a homomorphism of R to M. A left R-module M is called
I-P -injective if it is I-1-injective.
It is easy to see that direct sums and direct summands of I-n-injective modules are I-n-injective.
A left R-module M is n-injective if and only if M is R-n-injective, a left R-module M is J-injective
if and only if M is J-n-injective for every positive integer n. Follow [2], a ring R is said to be left
Soc-injective if every R-homomorphism from a semisimple submodule of RR to R extends to R.
Clearly, if Soc(RR) is finitely generated, then R is left Soc-injective if and only if RR is Soc(RR)-n-
injective for every positive integer n. We remark that J-P -injective modules are called JP -injective
in [22].
Theorem 2.1. Let M be a left R-module. Then the following statements are equivalent:
(1) M is I-n-injective.
(2) Ext1(R/T,M) = 0 for every n-generated left ideal T in I.
(3) rMnlRn(α) = αM for all α ∈ In.
(4) If x = (m1,m2, . . . ,mn)′ ∈Mn and α ∈ In satisfy lRn(α) ⊆ lRn(x), then x = αy for some
y ∈M.
(5) rMn(RnB ∩ lRn(α)) = rMn(B) + αM for all α ∈ In and B ∈ Rn×n.
(6) M is I-P -injective and rM (K ∩ L) = rM (K) + rM (L), where K and L are left ideals in I
such that K + L is n-generated.
(7) M is I-P -injective and rM (K ∩ L) = rM (K) + rM (L), where K and L are left ideals in I
such that K is cyclic and L is (n− 1)-generated.
(8) For each n-generated left ideal T in I and any f ∈ Hom(T,M), if (α, g) is the pushout of
(f, i) in the following diagram:
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 769
T
i−−−−→ R
f
y yg
M
α−−−−→ P
where i is the inclusion map, then there exists a homomorphism h : P →M such that hα = 1M .
Proof. (1)⇔ (2) and (8) ⇒ (1) are clear.
(1) ⇒ (3). Always αM ⊆ rMnlRn(α). If x ∈ rMnlRn(α), then the mapping f : Rnα → M ;
βα 7→ βx is a well-defined left R-homomorphism. Since M is I-n-injective and Rnα is an n-
generated left ideal in I, f can be extended to a homomorphism g of R to M. Let g(1) = y, then
x = αy ∈ αM. So rMnlRn(α) ⊆ αM. And thus rMnlRn(α) = αM.
(3) ⇒ (1). Let T =
∑n
i=1
Rai be an n-generated left ideal in I and f be a homomorphism
from T to M. Write ui = f(ai), i = 1, 2, . . . , n, u = (u1, u2, . . . , un)′, α = (a1, a2, . . . , an)′, then
u ∈ rMnlRn(α). By (3), there exists some x ∈ M such that u = αx. Now we define g : R → M ;
r 7→ rx, then g is a left R-homomorphism which extends f.
(3) ⇒ (4). If lRn(α) ⊆ lRn(x), where α ∈ In, x ∈ Mn, then x ∈ rMnlRn(x) ⊆ rMnlRn(α) =
= αM by (3). Thus (4) is proved.
(4) ⇒ (5). Let x ∈ rMn(RnB ∩ lRn(α)), then lRn(Bα) ⊆ lRn(Bx). By (4), Bx = Bαy for
some y ∈ M. Hence x − αy ∈ rMn(B), proving that rMn(RnB
⋂
lRn(α)) ⊆ rMn(B) + αM. The
other inclusion always holds.
(5) ⇒ (3). By taking B = E in (5).
(1) ⇒ (6). Clearly, M is I-P -injective and
rM (K) + rM (L) ⊆ rM (K ∩ L).
Conversely, let x ∈ rM (K ∩ L). Then f : K + L → M is well defined by f(k + l) = kx for
all k ∈ K and l ∈ L. Since M is I-n-injective, f = ·y for some y ∈ M. Hence, for all k ∈ K and
l ∈ L, we have ky = f(k) = kx and ly = f(l) = 0. Thus x − y ∈ rM (K) and y ∈ rM (L), so
x = (x− y) + y ∈ rM (K) + rM (L).
(6) ⇒ (7) is trivial.
(7) ⇒ (1). We proceed by induction on n. If n = 1, then (1) is clearly holds by hypothesis.
Suppose n > 1. Let T = Ra1 + Ra2 + . . . + Ran be an n-generated left ideal in I, T1 = Ra1
and T2 = Ra2 + . . . + Ran. Suppose f : T → M is a left R-homomorphism. Then f |T1 = ·y1
by hypothesis and f |T2 = ·y2 by induction hypothesis for some y1, y2 ∈ R. Thus y1 − y2 ∈
∈ rM (T1 ∩T2) = rM (T1) + rM (T2). So y1− y2 = z1 + z2 for some z1 ∈ rM (T1) and z2 ∈ rM (T2).
Let y = y1−z1 = y2 +z2. Then for any a ∈ T, let a = b1 + b2, b1 ∈ T1, b2 ∈ T2, we have b1z1 = 0,
b2z2 = 0. Hence f(a) = f(b1)+f(b2) = b1y1 +b2y2 = b1(y1−z1)+b2(y2 +z2) = b1y+b2y = ay.
So (1) follows.
(1) ⇒ (8). Without loss of generality, we may assume that P = (M ⊕ R)/W, where W =
= {f(a),−i(a) | a ∈ T}, g(r) = (0, r) +W, α(x) = (x, 0) +W for x ∈M and r ∈ R. Since M is
I-n-injective, there is ϕ ∈ HomR(R,M) such that ϕi = f. Define h[(x, r) +W ] = x+ ϕ(r) for all
(x, r) +W ∈ P. It is easy to check that h is well-defined and hα = 1M .
Theorem 2.1 is proved.
Corollary 2.1. Let M be a left R-module. Then the following statements are equivalent:
(1) M is n-injective.
(2) Ext1(R/T,M) = 0 for every n-generated left ideal T.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
770 ZHU ZHANMIN
(3) rMnlRn(α) = αM for all α ∈ Rn.
(4) If x = (m1,m2, . . . ,mn)′ ∈ Mn and α ∈ Rn satisfy lRn(α) ⊆ lRn(x), then x = αy for
some y ∈M.
(5) rMn(RnB ∩ lRn(α)) = rMn(B) + αM for all α ∈ Rn and B ∈ Rn×n.
(6) M is P -injective and rM (K ∩ L) = rM (K) + rM (L), where K and L are left ideals such
that K + L is n-generated.
(7) M is P -injective and rM (K ∩ L) = rM (K) + rM (L), where K and L are left ideals such
that K is cyclic and L is (n− 1)-generated.
(8) For each n-generated left ideal T and any f ∈ Hom(T,M), if (α, g) is the pushout of (f, i)
in the following diagram:
T
i−−−−→ R
f
y yg
M
α−−−−→ P
where i is the inclusion map, there exists a homomorphism h : P →M such that hα = 1M .
We note that the equivalence of (1), (3), (6), (7) in Corollary 2.1 appears in [6] (Corollaries 2.5
and 2.10).
Corollary 2.2. Let {Mα}α∈A be a family of right R-modules. Then
∏
α∈A
Mα is I-n-injective
if and only if each Mα is I-n-injective.
Proof. It follows from the isomorphism Ext1
(
N,
∏
α∈A
Mα
)
∼=
∏
α∈A
Ext1(N,Mα).
Recall that an element a ∈ R is called left I-semiregular [18] if there exists e2 = e ∈ Ra such
that a − ae ∈ I, and R is called left I-semiregular if every element is I-semiregular. A ring R is
called semiregular if R/J(R) is regular and idempotents lift modulo J(R). It is well known that
a ring R is semiregular if and only if it is left (equivalently right) J-semiregular [19]. Next, we
consider a case when I-n-injective modules are n-injective.
Theorem 2.2. Let R be a left I-semiregular ring. Then a left R-module M is n-injective if and
only if M is I-n-injective.
Proof. Necessity is clear. To prove sufficiency, let T be an n-generated left ideal and f : T →M
be a left R-homomorphism. Since R is left I-semiregular, by [18] (Theorem 1.2(2)), R = H ⊕ L,
where H ≤ T and T ∩ L ⊆ I. Hence R = T + L, T = H ⊕ (T ∩ L), and so T ∩ L is n-generated.
Since M is I-n-injective, there exists a homomorphism g : R → M such that g(x) = f(x) for all
x ∈ T ∩ L. Now let h : R → M ; r 7→ f(t) + g(l), where r = t + l, t ∈ T, l ∈ L. Then h is a
well-defined left R-homomorphism and h extends f.
Theorem 2.2 is proved.
Corollary 2.3. Let R be a left semiregular ring. Then:
(1) A left R-module M is P -injective if and only if M is JP -injective.
(2) A left R-module M is F -injective if and only if M is J-injective.
Theorem 2.3. Every pure submodule of an I-n-injective module is I-n-injective. In particular,
every pure submodule of an n-injective module is n-injective.
Proof. Let N be a pure submodule of an I-n-injective left R-module M. For any n-generated
left ideal T in I, we have the exact sequence
Hom(R/T,M)→ Hom(R/T,M/N)→ Ext1(R/T,N)→ Ext1(R/T,M) = 0.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 6
I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 771
Since R/T is finitely presented and N is pure in M, the sequence Hom(R/T,M)→ Hom(R/T,
M/N)→ 0 is exact. Hence Ext1(R/T,N) = 0, and so N is I-n-injective.
Theorem 2.3 is proved.
3. I-n-flat modules. Recall that a right R-module V is said to be n-flat [1, 9], if for every
n-generated left ideal T, the canonical map V ⊗ T → V ⊗ R is monic. 1-flat modules are called
P -flat by some authors such as Couchot [7]. A right R-module V is said to be J-flat [8], if for every
finitely generated left ideal T in J(R), the canonical map V ⊗ T → V ⊗R is monic. We extend the
concepts of n-flat modules and J-flat modules as follows.
Definition 3.1. A right R-module V is said to be I-n-flat, if for every n-generated left ideal T
in I, the canonical map V ⊗T → V ⊗R is monic. VR is said to be I-P -flat if it is I-1-flat. VR is said
to be I-flat if it is I-n-flat for every positive integer n.
It is easy to see that direct sums and direct summands and of I-n-flat modules are I-n-flat.
Theorem 3.1. For a right R-module V, the following statements are equivalent:
(1) V is I-n-flat.
(2) Tor1(V,R/T ) = 0 for every n-generated left ideal T in I .
(3) V + is I-n-injective .
(4) For every n-generated left ideal T in I, the map µT : V ⊗ T → V T ;
∑
vi ⊗ ai 7→
∑
viai
is a monomorphism.
(5) For all x ∈ V n, a ∈ In, if xa = 0, then exist positive integer m and y ∈ V m, C ∈ Rm×n,
such that Ca = 0 and x = yC.
Proof. (1)⇔ (2) follows from the exact sequence 0→ Tor1(V,R/T )→ V ⊗ T → V ⊗R.
(2)⇔ (3) follows from the isomorphism Tor1(M,R/T )+ ∼= Ext1(R/T,M+).
(1)⇔ (4) follows from the commutative diagram
V ⊗ T 1V ⊗iT−−−−→ V ⊗R
µT
y yσ
V T
iV T−−−−→ V
where σ is an isomorphism.
(4) ⇒ (5). Let x = (v1, v2, . . . , vn), a = (a1, a2, . . . , an)′, T =
∑n
j=1
Raj . Write ej be the
element in Rn with 1 in the jth position and 0’s in all other positions, j = 1, 2, . . . , n. Consider the
short exact sequence
0→ K
iK→ Rn
f→ T → 0
where f(ej) = aj for each j = 1, 2, . . . , n. Since xa = 0, by (4),
∑n
j=1
(vj⊗f(ej)) =
∑n
j=1
(vj⊗
⊗ aj) = 0 as an element in V ⊗R T. So in the exact sequence
V ⊗K 1V ⊗iK→ V ⊗Rn 1V ⊗f→ V ⊗ T → 0
we have
∑n
j=1
(vj ⊗ ej) ∈ Ker(1V ⊗ f) = Im(1V ⊗ iK). Thus there exist ui ∈ V, ki ∈ K,
i = 1, 2, . . . ,m, such that
∑n
j=1
(vj⊗ej) =
∑m
i=1
(ui⊗ki). Let ki =
∑n
i=1
cijej , j = 1, 2, . . . ,m.
Then
∑n
j=1
cijaj =
∑n
j=1
cijf(ej) = f(ki) = 0, i = 1, 2, . . . ,m. Write C = (cij)mn, then
Ca = 0.
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Moreover, this also gives
∑n
j=1
(vj⊗ ej) =
∑m
i=1
(ui⊗ki) =
∑m
i=1
(
ui ⊗
(∑n
j=1
cijej
))
=
=
∑n
j=1
((∑m
i=1
uicij
)
⊗ ej
)
. So vj =
∑m
i=1
uicij , j = 1, 2, . . . , n. Let y = (u1, u2, . . . , um),
then y ∈ V m and x = yC.
(5)⇒ (4). Let T =
∑n
j=1
Rbj be an n-generated left ideal in I and suppose ai =
∑n
j=1
rijbj ∈
∈ T, vi ∈ V with
∑k
i=1
viai = 0. Then
∑n
j=1
(∑k
i=1
virij
)
bj = 0. By (5), there exists elements
u1, . . . , um ∈ V and elements cij ∈ R, i = 1, . . . ,m, j = 1, . . . , n, such that
∑n
j=1
cijbj = 0,
i = 1, . . . ,m, and
∑m
i=1
uicij =
∑k
i=1
virij , j = 1, . . . , n. Thus,
∑k
i=1
vi ⊗ ai =
∑k
i=1
vi ⊗
⊗
(∑n
j=1
rijbj
)
=
∑n
j=1
(∑k
i=1
virij
)
⊗ bj =
∑n
j=1
(∑m
i=1
uicij
)
⊗ bj =
∑m
i=1
(
ui ⊗
⊗
∑n
j=1
cijbj
)
= 0. Thus (4) is proved.
Theorem 3.1 is proved.
Corollary 3.1. For a right R-module V, the following statements are equivalent:
(1) V is n-flat.
(2) Tor1(V,R/T ) = 0 for every n-generated left ideal T.
(3) V + is n-injective.
(4) For every n-generated left ideal T of R, the map µT : V ⊗T → V T ;
∑
vi⊗xi 7→
∑
vixi
is a monomorphism.
(5) For all x ∈ V n, a ∈ Rn, if xa = 0, then exist positive integer m and y ∈ V m, C ∈ Rm×n,
such that Ca = 0 and x = yC.
Corollary 3.2. Let R be a left I-semiregular ring. Then:
(1) A right R-module M is n-flat if and only if M is I-n-flat.
(2) A right R-module M is flat if and only if M is I-flat.
Proof. (1) follows from Corollary 3.1, Theorems 2.3 and 3.1.
(2) follows from (1).
Corollary 3.3. Let R be a left semiregular ring. Then:
(1) A right R-module M is n-flat if and only if M is J-n-flat.
(2) A right R-module M is flat if and only if M is J-flat.
We note that Corollary 3.3(2) improves the result of [8] (Proposition 2.17).
Corollary 3.4. Let {Mα}α∈A be a family of right R-modules and n be a positive integer. Then
(1)
⊕
α∈A
Mα is I-n-flat if and only if each Mα is I-n-flat.
(2)
∏
α∈A
Mα is I-n-injective if and only if each Mα is I-n-injective.
Proof. (1) follows from the isomorphism Tor1
(⊕
α∈A
Mα, N
)
∼=
⊕
α∈A
Tor1(Mα, N).
(2) follows from the isomorphism Ext1
(
N,
∏
α∈A
Mα
)
∼=
∏
α∈A
Ext1(N,Mα).
Remark 3.1. From Theorem 3.1, the I-n-flatness of VR can be characterized by the I-n-injectivity
of V +. On the other hand, by [5] (Lemma 2.7(1)), the sequence Tor1(V +,M)→ Ext1(M,V )+ → 0
is exact for all finitely presented left R-module M, so if V + is I-n-flat, then V is I-n-injective.
Theorem 3.2. Every pure submodule of an I-n-flat module is I-n-flat. In particular, pure sub-
modules of n-flat modules are n-flat.
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I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 773
Proof. Let A be a pure submodule of an I-n-flat right R-module B. Then the pure exact sequence
0 → A → B → B/A → 0 induces a split exact sequence 0 → (B/A)+ → B+ → A+ → 0. Since
B is I-n-flat, by Theorem 3.1, B+ is I-n-injective, and so A+ is I-n-injective. Thus A is I-n-flat
by Theorem 3.1 again.
Definition 3.2. Given a right R-module U with submodule U ′. If a = (a1, a2, . . . , an)′ ∈ Rn
and T =
∑n
i=1
Rai, then U ′ is called a-pure in U if the canonical map U ′ ⊗R R/T → U ⊗R R/T
is a monomorphism; U ′ is called I-n-pure in U if U ′ is a-pure in U for every a ∈ In. U ′ is called
I-P -pure in U if U ′ is I-1-pure in U.
Clearly, if U ′ is I-n-pure in U then U ′ is I-m-pure in U for every positive integer m ≤ n.
Theorem 3.3. Let U ′R ≤ UR and a = (a1, a2, . . . , an)′ ∈ Rn, T =
∑n
i=1
Rai. Then the
following statements are equivalent:
(1) U ′ is a-pure in U.
(2) The canonical map Tor1(U,R/T )→ Tor1(U/U ′, R/T ) is surjective.
(3) U ′ ∩ Una = (U ′)na.
(4) U ′ ∩ UT = U ′T .
(5) The canonical map HomR(Rn/aR,U)→ HomR(Rn/aR,U/U
′) is surjective.
(6) Every commutative diagram
aR
iaR−−−−→ Rn
f
y yg
U ′
iU′−−−−→ U
there exists h : Rn → U ′ with f = hiaR.
(7) The canonical map Ext1(Rn/aR,U
′) → Ext1(Rn/aR,U) is a monomorphism.
(8) lU
′
Un(a) = (U ′)n + lUn(a), where lU
′
Un(a) = {x ∈ Un|xa ∈ U ′}.
Proof. (1)⇔ (2). This follows from the exact sequence
Tor1(U,R/T )→ Tor1(U/U ′, R/T )→ U ′ ⊗R/T → U ⊗R/T.
(1) ⇒ (3). Suppose that x ∈ U ′ ∩ Una. Then there exists y = (y1, y2, . . . , yn) ∈ Un such
that x = ya, and so we have x ⊗
(
1 +
∑n
i=1
Rai
)
=
(∑n
i=1
yiai
)
⊗
(
1 +
∑n
i=1
Rai
)
=
=
∑n
i=1
(yi⊗ 0) = 0 in U ⊗
(
R/
∑n
i=1
Rai
)
. Since U ′ is a-pure in U, x⊗
(
1 +
∑n
i=1
Rai
)
= 0
in U ′⊗
(
R/
∑n
i=1
Rai
)
. Let ι :
∑n
i=1
Rai → R be the inclusion map and π : R→ R/
∑n
i=1
Rai
be the natural epimorphism. Then we have x⊗1 ∈ Ker(1U ′⊗π) = im (1U ′⊗ ι), it follows that there
exists x′i ∈ U ′, i = 1, 2, . . . , n, such that x ⊗ 1 =
∑n
i=1
x′i ⊗ ai =
(∑n
i=1
x′iai
)
⊗ 1 in U ′ ⊗ R,
and so x =
∑n
i=1
x′iai ∈ (U ′)na. But (U ′)na ⊆ U ′ ∩ Una, so U ′ ∩ Una = (U ′)na.
(3) ⇔ (4) is obvious.
(3) ⇒ (5). Consider the following diagram with exact rows:
0 −−−−→ aR
iaR−−−−→ Rn
π2−−−−→ Rn/aR −−−−→ 0yf
0 −−−−→ U ′
iU′−−−−→ U
π1−−−−→ U/U ′ −−−−→ 0
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where f ∈ HomR(Rn/aR,U/U
′). Since Rn is projective, there exist g ∈ HomR(Rn, U) and h ∈
∈ HomR(aR,U ′) such that the diagram commutes. Now let u = g(a), Then u = g(a) = h(a) ∈ U ′.
Note that u = (g(e1), g(e2), . . . , g(en))a ∈ Una, where ei ∈ Rn, with 1 in the ith position and 0’s in
all other positions. By (3), u ∈ (U ′)na. Therefore, u =
∑n
i=1
u′iai for some u′i ∈ U ′, i = 1, 2, . . . , n.
Define σ ∈ HomR(Rn, U
′) such that σ(ei) = u′i, i = 1, 2, . . . , n, then σiaR = h. Finally, we define
τ : Rn/aR→ U by τ(x+ aR) = g(x)− σ(x), then τ is a well-defined right R-homomorphism and
π1τ = f. Whence HomR(Rn/aR,U)→ HomR(Rn/aR,U/U
′) is surjective.
(5) ⇒ (3). Suppose that x ∈ U ′ ∩ Una. Then x = ya for some y = (y1, y2, . . . , yn) ∈ Un. Thus
we have the following commutative diagram with exact rows:
0 −−−−→ aR
iaR−−−−→ Rn
π2−−−−→ Rn/aR −−−−→ 0yf1 yf2
0 −−−−→ U ′
iU′−−−−→ U
π1−−−−→ U/U ′ −−−−→ 0
where f2 is defined by f2(ei) = yi, i = 1, 2, . . . , n and f1 = f2|aR. Define f3 : Rn/aR → U/U ′
by f3(z + aR) = π1f2(z). It is easy to see that f3 is well defined and f3π2 = π1f2. By hypothesis,
f3 = π1τ for some τ ∈ HomR(Rn/aR,U). Now we define σ : Rn → U ′ by σ(z) = f2(z)− τπ2(z).
Then σ ∈ HomR(Rn, U
′) and σ(a) = f2(a) since π2(a) = 0. Hence x = f2(a) = σ(a) =
= (σ(e1), σ(e2), . . . , σ(en))a ∈ (U ′)na. Therefore U ′ ∩ Una = (U ′)na.
(3) ⇒ (1). Suppose that
∑s
k=1
u′k ⊗
(
bk +
∑n
i=1
Rai
)
= 0 in U ⊗
(
R/
∑n
i=1
Rai
)
, where
u′k ∈ U ′, bk ∈ R, then
(∑s
k=1
u′kbk
)
⊗
(
1 +
∑n
i=1
Rai
)
= 0 in U ⊗
(
R/
∑n
i=1
Rai
)
. By
the exactness of the sequence U ⊗
(∑n
i=1
Rai
)
→ U ⊗ R → U ⊗
(
R/
∑n
i=1
Rai
)
→ 0, we
have that
∑s
k=1
u′kbk = xa for some x ∈ Un. By (3), there exists some y ∈ (U ′)n such that∑s
k=1
u′kbk = ya. Thus,
∑s
k=1
u′k ⊗
(
bk +
∑n
i=1
Rai
)
= ya ⊗
(
1 +
∑n
i=1
Rai
)
= 0 in U ′ ⊗
⊗
(
R/
∑n
i=1
Rai
)
.
(5) ⇔ (6). By diagram lemma (see [21, p. 53]).
(5) ⇔ (7). It follows from the exact sequence
HomR(Rn/aR,U)→ HomR(Rn/aR,U/U
′)→ Ext1(Rn/aR,U
′)→ Ext1(Rn/aR,U).
(5) ⇒ (8). It is sufficient to show that lU
′
Un(a) ⊆ (U ′)n + lUn(a). Let x = (x1, x2, . . . , xn) ∈
∈ lU
′
Un(a). Define f : Rn/aR→ U/U ′ via α+ aR 7→ xα+U ′, then f ∈ HomR(Rn/aR,U/U
′). By
(5), f = πg for some g ∈ HomR(Rn/aR,U), where π : U → U/U ′ is the natural epimorphism. Let
g(ei + aR) = yi, i = 1, 2, . . . , n, y = (y1, y2, . . . , yn). Then y ∈ lUn(a), xi + U ′ = f(ei + aR) =
= πg(ei + aR) = yi + U ′, and so xi − yi ∈ U ′, i = 1, 2, . . . , n, this implies that x − y ∈ (U ′)n.
Therefore, x = (x− y) + y ∈ (U ′)n + lUn(a).
(8) ⇒ (6). Let x = (g(e1), g(e2), . . . , g(en)). Then xa = g(a) = f(a) ∈ U ′, so x ∈ lU
′
Un(a). By
(8), x = y + z for some y ∈ (U ′)n and z ∈ lUn(a). Now we define h : Rn → U ′; b 7→ yb, then
h(a) = ya = xa = f(a). And thus f = hiaR.
Theorem 3.3 is proved.
Let M be a right R-module, K be a submodule of M and X a subset of M, then we write
X/K = {x+K|x ∈ X}.
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Corollary 3.5. Suppose that E,F and G are right R-modules such that E ⊆ F ⊆ G, and
a ∈ Rn. Then:
(1) If E is a-pure in F and F is a-pure in G, then E is a-pure in G.
(2) If E is a-pure in G, then E is a-pure in F .
(3) If F is a-pure in G, then F/E is a-pure in G/E.
(4) If E is a-pure in G and F/E is a-pure in G/E, then F is a-pure in G.
Proof. (1). Since E is a-pure in F and F is a-pure in G, we have F ∩ Gna = Fna and
E ∩ Fna = Ena. Thus, E ∩Gna = E ∩ (F ∩Gna) = E ∩ Fna = Ena, and therefore E is a-pure
in G.
(2) Since E is a-pure in G, E ∩ Gna = Ena. Note that E ∩ Gna ⊇ E ∩ Fna ⊇ Ena, we get
that E ∩ Fna = Ena, and (2) follows.
(3) Since F is a-pure in G, F ∩ Gna = Fna, and so (F/E) ∩ (G/E)na = (F ∩ Gna)/E =
= (Fna)/E = (F/E)na. This follows that F/E is a-pure in G/E.
(4) By hypothesis, we have (F/E)∩ (G/E)na = (F/E)na, i.e., (F ∩Gna)/E = (Fna)/E, and
E ∩ Gna = Ena. For any f ∈ F ∩ Gna, write f = ga, where g ∈ Gn. Then there exists f1 ∈ Fn
such that (g − f1)a = ga− f1a = f − f1a ∈ E ∩Gna = Ena, so f − f1a = ea for some e ∈ En.
This implies that f = f1a+ ea = (f1 + e)a ∈ Fna, and hence F is a-pure in G.
Corollary 3.6. Let U ′R ≤ UR and a ∈ R. Then the following statements are equivalent:
(1) U ′ is a-pure in U.
(2) The canonical map Tor1(U,R/Ra)→ Tor1(U/U ′, R/Ra) is surjective.
(3) U ′ ∩ Ua = U ′a.
(4) The canonical map HomR(R/aR,U)→ HomR(R/aR,U/U ′) is surjective.
(5) Every commutative diagram
aR
iaR−−−−→ R
f
y yg
U ′
iU′−−−−→ U
there exists h : R→ U ′ with f = hiaR.
(6) The canonical map Ext1(R/aR,U ′)→ Ext1(R/aR,U) is a monomorphism.
(7) lU
′
U (a) = U ′ + lU (a), where lU
′
U (a) = {x ∈ U | xa ∈ U ′}.
Corollary 3.7. Let U be an n-generated right R-module with submodule U ′. If U ′ is I-n-pure
in U, then U ′ is I-m-pure in U for each positive integer m. In particular, if a right ideal T of R is
I-P -pure in R, then it is I-m-pure in R for each positive integer m.
Proof. For any a ∈ Im, if x ∈ U ′ ∩ Uma, then x = (x1, x2, . . . , xm)a, where each xi ∈ U.
Suppose that u1, u2, . . . , un is a generating set of U. Then (x1, x2, . . . , xm) = (u1, u2, . . . , un)C for
some C ∈ Rn×m, and so x = (u1, u2, . . . , un)(Ca) ∈ U ′ ∩ Un(Ca). Since U ′ is I-n-pure in U, by
Theorem 3.3, x ∈ (U ′)n(Ca) = ((U ′)nC)a ⊆ (U ′)ma. Thus U ′ ∩ Uma = (U ′)ma and therefore U ′
is I-m-pure in U.
Proposition 3.1. Let U ′R ≤ UR.
(1) If U/U ′ is I-n-flat, then U ′ is I-n-pure in U .
(2) If U ′ is I-n-pure in U and U is I-n-flat, then U/U ′ is I-n-flat.
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Proof. It follows from the exact sequence
Tor1(U,R/T )→ Tor1(U/U ′, R/T )→ U ′ ⊗R/T → U ⊗R/T
and Theorem 3.1(2).
Theorem 3.4. n-Generated I-n-flat module is I-flat.
Proof. Suppose V is an n-generated I-n-flat module, there exists an exact sequence 0 → K →
→ F → V → 0 with F free and rank(F ) = n. Then K is I-n-pure in F by Proposition 3.1(1)
and hence I-m-pure for every positive integer m by Corollary 3.7. So, by Proposition 3.1(2), V is
I-m-flat for every positive integer m. Hence, V is I-flat.
Theorem 3.4 is proved.
Corollary 3.8. (1) n-Generated n-flat module is flat.
(2) I-P -flat cyclic module is I-flat.
4. I-n-coherent rings.
Definition 4.1. A ring R is called left I-n-coherent if every n-generated left ideal in I is finitely
presented.
Clearly, a ring R is left n-coherent if and only if R is left R-n-coherent.
Lemma 4.1. Let a ∈ Rn. Then lRn(a) ∼= P ∗, where P = Rn/aR.
Proof. This is a corollary of [23] (Lemma 5.3).
Theorem 4.1. The following statements are equivalent for a ring R:
(1) R is left I-n-coherent.
(2) If 0→ K
f→ Rn
g→ I is an exact sequence of left R-modules, then K is finitely generated.
(3) lRn(a) is a finitely generated submodule of Rn for any a ∈ In.
(4) For any a ∈ In, (Rn/aR)∗ is finitely generated.
Proof. (1)⇒ (2). Since R is left I-n-coherent and Im(g) is an n-generated left ideal in I, Im(g)
is finitely presented. Noting that the sequence 0→ Ker(g)→ Rn → Im(g)→ 0 is exact, so Ker(g)
is finitely generated. Thus K ∼= Im(f) = Ker(g) is finitely generated.
(2) ⇒ (3). Let a = (a1, . . . , an)′. Then we have an exact sequence of left R-modules 0 →
→ lRn(a)→ Rn
g→ I, where g(r1, . . . , rn) =
∑n
i=1
riai. By (2), lRn(a) is a finitely generated left
R-module.
(3) ⇒ (1) is obvious. (3)⇔ (4) follows from Lemma 4.1.
Theorem 4.1 is proved.
Let F be a class of right R-modules and M a right R-module. Following [10], we say that
a homomorphism ϕ : M → F where F ∈ F is an F-preenvelope of M if for any morphism
f : M → F ′ with F ′ ∈ F , there is a g : F → F ′ such that gϕ = f. An F-preenvelope ϕ : M → F
is said to be an F-envelope if every endomorphism g : F → F such that gϕ = ϕ is an isomorphism.
Dually, we have the definitions of an F-precover and an F-cover. F-envelopes (F-covers) may not
exist in general, but if they exist, they are unique up to isomorphism.
Theorem 4.2. The following statements are equivalent for a ring R:
(1) R is left I-n-coherent.
(2) lim−→Ext1(R/T,Mα) ∼= Ext1(R/T, lim−→Mα) for every n-generated left ideal T in I and
direct system (Mα)α∈A of left R-modules.
(3) Tor1
(∏
Nα, R/T
)
∼=
∏
Tor1(Nα, R/T ) for any family {Nα} of right R-modules and
any n-generated left ideal T in I.
(4) Any direct product of copies of RR is I-n-flat.
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I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 777
(5) Any direct product of I-n-flat right R-modules is I-n-flat.
(6) Any direct limit of I-n-injective left R-modules is I-n-injective.
(7) Any direct limit of injective left R-modules is I-n-injective.
(8) A left R-module M is I-n-injective if and only if M+ is I-n-flat.
(9) A left R-module M is I-n-injective if and only if M++ is I-n-injective.
(10) A right R-module M is I-n-flat if and only if M++ is I-n-flat.
(11) For any ring S, Tor1(HomS(B,C), R/T ) ∼= HomS(Ext1(R/T,B), C) for the situation
(R(R/T ),RBS , CS) with T n-generated left ideal in I and CS injective.
(12) Every right R-module has an I-n-flat preenvelope.
(13) For any U ∈ In, U(R) is a finitely generated left ideal, where U(R) = {r ∈ R : (r, r2, . . .
. . . , rn)U = 0 for some r2, . . . , rn ∈ R}.
Proof. (1) ⇒ (2) follows from [5] (Lemma 2.9(2)).
(1)⇒ (3) follows from [5] (Lemma 2.10(2)).
(2)⇒ (6) ⇒ (7); (3)⇒ (5) ⇒ (4) are trivial.
(7) ⇒ (1). Let T be an n-generated left ideal in I and (Mα)α∈A a direct system of injective left
R-modules (with A directed). Then lim−→Mα is I-n-injective by (7), and so Ext1(R/T, lim−→Mα) = 0.
Thus we have a commutative diagram with exact rows:
lim−→Hom(R/T,Mα) −−−−→ lim−→Hom(R,Mα) −−−−→ lim−→Hom(T,Mα) −−−−→ 0yf yg yh
Hom(R/T, lim−→Mα) −−−−→ Hom(R, lim−→Mα) −−−−→ Hom(T, lim−→Mα) −−−−→ 0.
Since f and g are isomorphism by [21] (25.4(d)), h is an isomorphism by the Five lemma. So T is
finitely presented by [21] (25.4(e)) again. Hence R is left I-n-coherent.
(4) ⇒ (1). Let T be an n-generated left ideal in I. By (4), Tor1(ΠR,R/T ) = 0. Thus we have
a commutative diagram with exact rows:
0 −−−−→ (ΠR)⊗ T −−−−→ (ΠR)⊗R −−−−→ (ΠR)⊗R/T −−−−→ 0yf1 yf2 yf3
0 −−−−→ ΠT −−−−→ ΠR −−−−→ Π(R/T ) −−−−→ 0
Since f3 and f2 are isomorphism by [10] (Theorem 3.2.22), f1 is an isomorphism by the Five lemma.
So T is finitely presented by [10] (Theorem 3.2.22) again. Hence R is left I-n-coherent.
(5)⇒ (12). Let N be any right R-module. By [10] (Lemma 5.3.12), there is a cardinal number ℵα
dependent on Card(N) and Card(R) such that for any homomorphism f : N → F with F I-n-flat,
there is a pure submodule S of F such that f(N) ⊆ S and CardS ≤ ℵα. Thus f has a factorization
N → S → F with S I-n-flat by Theorem 3.2. Now let {ϕβ}β∈B be all such homomorphisms
ϕβ : N → Sβ with CardSβ ≤ ℵα and Sβ I-n-flat. Then any homomorphism N → F with F
I-n-flat has a factorization N → Si → F for some i ∈ B. Thus the homomorphism N → Πβ∈BSβ
induced by all ϕβ is an I-n-flat preenvelope since Πβ∈BSβ is I-n-flat by (5).
(12)⇒ (5) follows from [4] (Lemma 1).
(1) ⇒ (11). For any n-generated left ideal T in I, since R is left I-n-coherent, R/T is 2-
presented. And so (11) follows from [5] (Lemma 2.7(2)).
(11) ⇒ (8). Let S = Z, C = Q/Z and B = M. Then Tor1(M+, R/T ) ∼= Ext1(R/T,M)+ for
any n-generated left ideal T in I by (11), and hence (8) holds.
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(8) ⇒ (9). Let M be a left R-module. If M is I-n-injective, then M+ is I-n-flat by (8), and so
M++ is I-n-injective by Theorem 3.1. Conversely, if M++ is I-n-injective, then M, being a pure
submodule of M++ (see [20, p. 48], Exercise 41), is I-n-injective by Theorem 2.3.
(9) ⇒ (10). If M is an I-n-flat right R-module, then M+ is an I-n-injective left R-module by
Theorem 3.1, and so M+++ is I-n-injective by (9). Thus M++ is I-n-flat by Theorem 3.1 again.
Conversely, if M++ is I-n-flat, then M is I-n-flat by Theorem 3.2 as M is a pure submodule
of M++.
(10) ⇒ (5). Let {Nα}α∈A be a family of I-n-flat right R-modules. Then by Corollary 3.4(1),⊕
α∈A
Nα is I-n-flat, and so
(∏
α∈A
N+
α
)+ ∼=
(⊕
α∈A
Nα
)++
is I-n-flat by (10). Since
⊕
α∈A
N+
α is a
pure submodule of
∏
α∈A
N+
α by [3] (Lemma 1(1)),
(∏
α∈A
N+
α
)+
→
(⊕
α∈AN
+
α
)+ → 0 split,
and hence
(⊕
α∈A
N+
α
)+
is I-n-flat. Thus
∏
α∈A
N++
α
∼=
(⊕
α∈A
N+
α
)+
is I-n-flat. Since
∏
α∈A
Nα
is a pure submodule of
∏
α∈A
N++
α by [3] (Lemma 1(2)),
∏
α∈A
Nα is I-n-flat by Theorem 3.2.
(1) ⇒ (13). Let U = (u1, u2, . . . , un)′ ∈ In. Write T1 = Ru1 + Ru2 + . . . + Run and T2 =
= Ru2 + . . .+Run. Then R/U(R) ∼= T1/T2. By (1), T1 is finitely presented, and so T1/T2 is finitely
presented. Therefore U(R) is finitely generated.
(13) ⇒ (1). Let T1 = Ru1 + Ru2 + . . . + Run be an n-generated left ideal in I. Let T2 =
= Ru2 + . . . + Run, T3 = Ru3 + . . . + Run, . . . , Tn = Run. Then T1/T2
∼= R/U(R) is finitely
presented by (13). Similarly, T2/T3, . . . , Tn−1/Tn, Tn are finitely presented. Hence T1 is finitely
presented, and (1) follows.
Theorem 4.2 is proved.
Corollary 4.1. The following statements are equivalent for a ring R:
(1) R is left n-coherent.
(2) lim−→ Ext1(R/T,Mα) ∼= Ext1(R/T, lim−→Mα) for every n-generated left ideal T and direct
system (Mα)α∈A of left R-modules.
(3) Tor1(
∏
Nα, R/T ) ∼=
∏
Tor1(Nα, R/T ) for any family {Nα} of right R-modules and any
n-generated left ideal T.
(4) Any direct product of copies of RR is n-flat.
(5) Any direct product of n-flat right R-modules is n-flat.
(6) Any direct limit of n-injective left R-modules is n-injective.
(7) Any direct limit of injective left R-modules is n-injective.
(8) A left R-module M is n-injective if and only if M+ is n-flat.
(9) A left R-module M is n-injective if and only if M++ is n-injective.
(10) A right R-module M is n-flat if and only if M++ is n-flat.
(11) For any ring S, Tor1(HomS(B,C), R/T ) ∼= HomS(Ext1(R/T,B), C) for the situation
(R(R/T ),RBS , CS) with T n-generated left ideal and CS injective.
(12) Every right R-module has an n-flat preenvelope.
(13) For any U ∈ Rn, U(R) is a finitely generated left ideal, where
U(R) = {r ∈ R : (r, r2, . . . , rn)U = 0 for some r2, . . . , rn ∈ R}.
Corollary 4.2. The following statements are equivalent for a ring R:
(1) R is left coherent.
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I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 779
(2) lim−→ Ext1(R/T,Mα) ∼= Ext1(R/T, lim−→Mα) for every finitely generated left ideal T and
direct system (Mα)α∈A of left R-modules.
(3) Tor1
(∏
Nα, R/T
)
∼=
∏
Tor1(Nα, R/T ) for any family {Nα} of right R-modules and
any finitely generated left ideal T.
(4) Any direct product of copies of RR is flat.
(5) Any direct product of flat right R-modules is flat.
(6) Any direct limit of F -injective left R-modules is F -injective.
(7) Any direct limit of injective left R-modules is F -injective.
(8) A left R-module M is F -injective if and only if M+ is flat.
(9) A left R-module M is F -injective if and only if M++ is F -injective.
(10) A right R-module M is flat if and only if M++ is flat.
(11) For any ring S, Tor1(HomS(B,C), R/T ) ∼= HomS(Ext1(R/T,B), C) for the situation
(R(R/T ),RBS , CS) with T finitely generated left ideal and CS injective.
(12) For any positive integer n and any U ∈ Rn, U(R) is a finitely generated left ideal, where
U(R) = {r ∈ R : (r, r2, . . . , rn)U = 0 for some r2, . . . , rn ∈ R}.
(13) Every right R-module has a flat preenvelope.
Proof. The equivalence of (1) – (12) is a consequence of Corollary 4.1. The proof of (5) ⇔ (13)
is similar to that of (5)⇔ (12) in the proof of Theorem 4.2.
Corollary 4.3. Let R be a left I-n-coherent ring. Then every left R-module has an I-n-injective
cover.
Proof. Let 0 → A → B → C → 0 be a pure exact sequence of left R-modules with B I-n-
injective. Then 0 → C+ → B+ → A+ → 0 is split. Since R is left I-n-coherent, B+ is I-n-flat
by Theorem 4.2, so C+ is I-n-flat, and hence C is I-n-injective by Remark 3.1. Thus, the class of
I-n-injective modules is closed under pure quotients. By [12] (Theorem 2.5), every left R-module
has an I-n-injective cover.
Corollary 4.4. LetR be a left n-coherent ring. Then every leftR-module has an n-injective cover.
Proposition 4.1. Let R be a left coherent ring. Then every left R-module has a F -injective
cover.
Proof. It is similar to the proof of Corollary 4.3.
Corollary 4.5. The following are equivalent for a left I-n-coherent ring R:
(1) Every I-n-flat right R-module is n-flat.
(2) Every I-n-injective left R-module is n-injective.
In this case, R is left n-coherent.
Proof. (1) ⇒ (2). Let M be any I-n-injective left R-module. Then M+ is I-n-flat by Theo-
rem 4.2, and so M+ is n-flat by (1). Thus M++ is n-injective by Corollary 3.1. Since M is a pure
submodule of M++, and pure submodule of an n-injective module is n-injective by Theorem 2.3, so
M is n-injective.
(2) ⇒ (1). Let M be any I-n-flat right R-module. Then M+ is I-n-injective left R-module by
Theorem 3.1, and so M+ is n-injective by (2). Thus M is n-flat by Corollary 3.1.
In this case, any direct product of n-flat right R-modules is n-flat by Theorem 4.2, and so R is
left n-coherent by Corollary 4.1.
Corollary 4.6. Left I-semiregular left I-n-coherent ring is left n-coherent.
Proof. By Corollaries 3.2(1) and 4.5.
Corollary 4.7. Semiregular left J-coherent ring is left coherent.
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Proposition 4.2. The following statements are equivalent for a left I-n-coherent ring R:
(1) RR is I-n-injective.
(2) Every right R-module has a monic I-n-flat preenvelope.
(3) Every left R-module has an epic I-n-injective cover.
(4) Every injective right R-module is I-n-flat.
Proof. (1)⇒ (2). LetM be any right R-module. ThenM has an I-n-flat preenvelope f : M → F
by Theorem 4.2. Since (RR)+ is a cogenerator, there exists an exact sequence 0→M
g→
∏
(RR)+.
Since RR is I-n-injective, by Theorem 4.2,
∏
(RR)+ is I-n-flat, and so there exists a right R-
homomorphism h : F →
∏
(RR)+ such that g = hf, which shows that f is monic.
(2) ⇒ (4). Assume (2). Then for every injective right R-module E, E has a monic I-n-flat
preenvelope F, so E is isomorphism to a direct summand of F, and thus E is I-n-flat.
(4) ⇒ (1). Since (RR)+ is injective, by (4), it is I-n-flat. Thus RR is I-n-injective by Theo-
rem 4.2.
(1) ⇒ (3). Let M be a left R-module. Then M has an I-n-injective cover ϕ : C → M by
Corollary 4.3. On the other hand, there is an exact sequence F
α→ M → 0 with F free. Since F is
I-n-injective by (1), there exists a homomorphism β : F → C such that α = ϕβ. This follows that
ϕ is epic.
(3) ⇒ (1). Let f : N → RR be an epic I-n-injective cover. Then the projectivity of RR implies
that RR is isomorphism to a direct summand of N, and so RR is I-n-injective.
Proposition 4.2 is proved.
Corollary 4.8. The following statements are equivalent for a left n-coherent ring R:
(1) RR is n-injective.
(2) Every right R-module has a monic n-flat preenvelope.
(3) Every left R-module has an epic n-injective cover.
(4) Every injective right R-module is n-flat.
Proposition 4.3. The following statements are equivalent for a left coherent ring R:
(1) RR is F -injective.
(2) Every right R-module has a monic flat preenvelope.
(3) Every left R-module has an epic F -injective cover.
(4) Every injective right R-module is flat.
Proof. It is similar to the proof of Proposition 4.2.
5. I-n-semihereditary rings.
Definition 5.1. A ring R is called left I-n-semihereditary if every n-generated left ideal in I
is projective. A ring R is called left I-semihereditary if every finitely generated left ideal in I is
projective. A ring R is called left IPP if every principal left ideal in I is projective. A ring R is called
left JPP if every principal left ideal in J is projective.
Recall that a ring R is called left PP [13] if every principal left ideal is projective. It is easy
to see that a ring R is left PP if and only if R is left R-1-semihereditary, a ring R is left JPP if
and only if R is left J-1-semihereditary, a ring R is left n-semihereditary if and only if R is left
R-n-semihereditary, a ring R is left J-semihereditary if and only if R is left J-n-semihereditary for
every positive integer n.
Theorem 5.1. The following statements are equivalent for a ring R:
(1) R is a left I-n-semihereditary ring.
(2) R is left I-n-coherent and submodules of I-n-flat right R-modules are I-n-flat.
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I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 781
(3) R is left I-n-coherent and every right ideal is I-n-flat.
(4) R is left I-n-coherent and every finitely generated right ideal is I-n-flat.
(5) Every quotient module of an I-n-injective left R-module is I-n-injective.
(6) Every quotient module of an injective left R-module is I-n-injective.
(7) Every left R-module has a monic I-n-injective cover.
(8) Every right R-module has an epic I-n-flat envelope.
(9) For every left R-module A, the sum of an arbitrary family of I-n-injective submodules of A
is I-n-injective.
Proof. (2)⇒ (3) ⇒ (4), and (5)⇒ (6) are trivial.
(1)⇒ (2). R is clearly left I-n-coherent. Let A be a submodule of an I-n-flat right R-module B
and T an n-generated left ideal in I. Then T is projective by (1) and hence flat. Then the exactness
of 0 = Tor2(B/A,R)→ Tor2(B/A,R/T )→ Tor1(B/A, T ) = 0 implies that Tor2(B/A,R/T ) =
= 0. And thus from the exactness of the sequence 0 = Tor2(B/A,R/T ) → Tor1(A,R/T ) →
→ Tor1(B,R/T ) = 0 we have Tor1(A,R/T ) = 0, this follows that A is I-n-flat.
(4) ⇒ (1). Let T be an n-generated left ideal in I. Then for any finitely generated right
ideal K of R, the exact sequence 0 → K → R → R/K → 0 implies the exact sequence
0 → Tor2(R/K,R/T ) → Tor1(K,R/T ) = 0 since K is I-n-flat. So Tor2(R/K,R/T ) = 0,
and hence we obtain an exact sequence 0 = Tor2(R/K,R/T ) → Tor1(R/K, T ) → 0. Thus,
Tor1(R/K, T ) = 0, and so T is a finitely presented flat left R-module. Therefore, T is projective.
(1) ⇒ (5). Let M be an I-n-injective left R-module and N be a submodule of M. Then for
any n-generated left ideal T in I, since T is projective, the exact sequence 0 = Ext1(T,N) →
→ Ext2(R/T,N) → Ext2(R,N) = 0 implies that Ext2(R/T,N) = 0. Thus the exact sequence
0 = Ext1(R/T,M)→ Ext1(R/T,M/N)→ Ext2(R/T,N) = 0 implies that Ext1(R/T,M/N) =
= 0. Consequently, M/N is I-n-injective.
(6) ⇒ (1). Let T be an n-generated left ideal in I. Then for any left R-module M, by hy-
pothesis, E(M)/M is I-n-injective, and so Ext1(R/T,E(M)/M) = 0. Thus, the exactness of the
sequence 0 = Ext1(R/T,E(M)/M) → Ext2(R/T,M) → Ext2(R/T,E(M)) = 0 implies that
Ext2(R/T,M) = 0. Hence, the exactness of the sequence 0 = Ext1(R,M) → Ext1(T,M) →
→ Ext2(R/T,M) = 0 implies that Ext1(T,M) = 0, this shows that T is projective, as required.
(2), (5) ⇒ (7). Since R is left I-n-coherent by (2), for any left R-module M, there is an I-n-
injective cover f : E →M by Corollary 4.3. Note that im(f) is I-n-injective by (5), and f : E →M
is an I-n-injective precover, so for the inclusion map i : im(f) → M, there is a homomorphism
g : im(f) → E such that i = fg. Hence f = f(gf). Observing that f : E → M is an I-n-injective
cover and gf is an endomorphism of E, so gf is an automorphisms of E, and hence f : E → M is
a monic I-n-injective cover.
(7)⇒ (5). Let M be an I-n-injective left R-module and N be a submodule of M. By (7), M/N
has a monic I-n-injective cover f : E → M/N. Let π : M → M/N be the natural epimorphism.
Then there exists a homomorphism g : M → E such that π = fg. Thus f is an isomorphism, and
whence M/N ∼= E is I-n-injective.
(2) ⇔ (8). By Theorem 4.2 and [4] (Theorem 2).
(5) ⇒ (9). Let A be a left R-module and {Aγ | γ ∈ Γ} be an arbitrary family of I-n-injective
submodules of A . Since the direct sum of I-n-injective modules is I-n-injective and
∑
γ∈Γ
Aγ is a
homomorphic image of ⊕γ∈ΓAγ , by (5),
∑
γ∈Γ
Aγ is I-n-injective.
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(9) ⇒ (6). Let E be an injective left R-module and K ≤ E. Take E1 = E2 = E, N = E1 ⊕
⊕ E2, D = {(x,−x) | x ∈ K}. Define f1 : E1 → N/D by x1 7→ (x1, 0) + D, f2 : E2 → N/D by
x2 7→ (0, x2) + D and writeEi = fi(Ei), i = 1, 2. Then Ei ∼= Ei is injective, i = 1, 2, and hence
N/D = E1 + E2 is I-n-injective. By the injectivity of Ei, (N/D)/Ei is isomorphic to a summand
of N/D and thus it is I-n-injective.
Theorem 5.1 is proved.
Corollary 5.1. The following statements are equivalent for a ring R:
(1) R is a left n-semihereditary ring.
(2) R is left n-coherent and submodules of n-flat right R-modules are n-flat.
(3) R is left n-coherent and every right ideal is n-flat.
(4) R is left n-coherent and every finitely generated right ideal is n-flat.
(5) Every quotient module of an n-injective left R-module is n-injective.
(6) Every quotient module of an injective left R-module is n-injective.
(7) Every left R-module has a monic n-injective cover.
(8) Every right R-module has an epic n-flat envelope.
(9) For every left R-module A, the sum of an arbitrary family of n-injective submodules of A is
n-injective.
Recall that a ring R is called left P -coherent [15] if it is left 1-coherent.
Corollary 5.2. The following statements are equivalent for a ring R:
(1) R is a left PP ring.
(2) R is left P -coherent and submodules of P -flat right R-modules are P -flat.
(3) R is left P -coherent and every right ideal is P -flat.
(4) R is left P -coherent and every finitely generated right ideal is P -flat.
(5) Every quotient module of a P -injective left R-module is P -injective.
(6) Every quotient module of an injective left R-module is P -injective.
(7) Every left R-module has a monic P -injective cover.
(8) Every right R-module has an epic P -flat envelope.
(9) For every left R-module A, the sum of an arbitrary family of P -injective submodules of A is
P -injective.
Corollary 5.3. The following statements are equivalent for a ring R:
(1) R is a left JPP ring.
(2) R is left J-P -coherent and submodules of J-P -flat right R-modules are J-P -flat.
(3) R is left J-P -coherent and every right ideal is J-P -flat.
(4) R is left J-P -coherent and every finitely generated right ideal is J-P -flat.
(5) Every quotient module of a J-P -injective left R-module is J-P -injective.
(6) Every quotient module of an injective left R-module is J-P -injective.
(7) Every left R-module has a monic J-P -injective cover.
(8) Every right R-module has an epic J-P -flat envelope.
(9) For every left R-module A, the sum of an arbitrary family of J-P -injective submodules of A
is J-P -injective.
Proposition 5.1. Let R be an left I-semiregular ring. Then:
(1) R is left n-semihereditary if and only if it is left I-n-semihereditary.
(2) R is left semihereditary if and only if it is left I-semihereditary.
(3) R is left PP if and only if it is left IPP.
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I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 783
Proof. (1). We need only to prove the sufficiency. Suppose R is left I-n-semihereditary, then
by Theorem 5.1, every quotient module of an injective left R-module is I-n-injective. Since R
is left I-semiregular, every I-n-injective left R-module is n-injective by Theorem 2.2. So every
quotient module of an injective left R-module is n-injective, and hence R is left n-semihereditary by
Corollary 5.1.
(2), (3) follows from (1).
Proposition 5.1 is proved.
From Proposition 5.1, we have immediately the following results.
Corollary 5.4. Let R be a semiregular ring. Then:
(1) R is left n-semihereditary if and only if it is left J-n-semihereditary.
(2) R is left semihereditary if and only if it is left J-semihereditary.
(3) R is left PP if and only if it is left JPP.
6. I-P -injective rings and I-regular rings. In this section we extend the concept of regular
rings to I-regular rings, give some characterizations of I-regular rings and I-P -injective modules,
and give some properties of left I-P -injective rings.
Definition 6.1. A ring R is called I-regular if every element in I is regular.
Clearly, every ring is 0-regular, R is semiprimitive if and only if R is J-regular, R is regular if
and only R is R-regular.
We call a module M is absolutely I-P -pure if M is I-P -pure in every module containing M.
Theorem 6.1. Let M be a left R-module. Then the following statements are equivalent:
(1) M is I-P -injective.
(2) Ext1(R/Ra,M) = 0 for all a ∈ I.
(3) rM lR(a) = aM for all a ∈ I.
(4) lR(a) ⊆ lR(x), where a ∈ I, x ∈M, implies x ∈ aM.
(5) rM (Rb ∩ lR(a)) = rM (b) + aM for all a ∈ I and b ∈ R.
(6) If γ : Ra→M, a ∈ I, is R-linear, then γ(a) ∈ aM.
(7) M is absolutely I-P -pure.
(8) M is I-P -pure in its injective envelope E(M).
(9) M is an I-P -pure submodule of an I-P -injective module.
(10) For each a ∈ I and any f ∈ Hom(Ra,M), if (α, g) is the pushout of (f, i) in the following
diagram:
aR
i−−−−→ R
f
y yg
M
α−−−−→ P
where i is the inclusion map, then there exists a homomorphism h : P →M such that hα = 1M .
Proof. (1) ⇔ (2) ⇔ (3) ⇔ (4) ⇔ (5) ⇔ (10) are follows from Theorem 2.1. (7) ⇒ (8) ⇒ (9)
are clear.
(4) ⇒ (6). Let γ : Ra → M be R-linear, where a ∈ I. Then lR(a) ⊆ lR(γ(a)). By (4),
γ(a) ∈ aM.
(6) ⇒ (1). Let γ : Ra → M be R-linear, where a ∈ I. By (6), write γ(a) = am, m ∈ M. Then
γ = ·m, proving (1).
(2) ⇒ (7). By Theorem 3.3(5).
(9) ⇒ (2). Let M be an I-P -pure submodule of an I-P -injective module N. Then (2) follows
from the the exact sequence
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784 ZHU ZHANMIN
HomR(R/Ra,N)→ HomR(R/Ra,N/M)→ Ext1
R(R/Ra,M)→ 0
and Theorem 3.3(5).
Theorem 6.1 is proved.
Corollary 6.1. Let R = I1 ⊕ I2, where I1, I2 are ideals of R. Then R is left P -injective if and
only if RR is I1-P -injective and I2-P -injective.
Proof. We need only to prove the sufficiency. Let a = a1+a2 ∈ R, where a1 ∈ I1, a2 ∈ I2. Then
by routine computations, we have rRlR(a1) = rI1lI1(a1), rRlR(a2) = rI2lI2(a2), rRlR(a1 + a2) =
= rI1lI1(a1) + rI2lI2(a2), a1R + a2R = (a1 + a2)R. Since R is left I1-P -injective and left I2-P -
injective, rRlR(a1) = a1R, rRlR(a2) = a2R. Hence, rRlR(a) = aR, which shows that R is left
P -injective.
Proposition 6.1. Let R be a left I-P -injective ring. Then:
(1) Every left ideal in I that is isomorphic to a direct summand of RR is itself a direct summand
of RR.
(2) If Re ∩Rf = 0, e2 = e ∈ R, f2 = f ∈ I, then Re⊕Rf = Rg for some g2 = g.
(3) If Rk is a simple left ideal in I, then kR is a simple right ideal.
(4) Soc(RI) ⊆ Soc(IR).
Proof. (1). If T is a left ideal in I and T ∼= Re, where e2 = e ∈ R, then T = Ra for some
a ∈ T and T is projective. Hence lR(a) ⊆⊕ RR, say lR(a) = Rf, where f2 = f ∈ R. Then
aR = rRlR(a) = (1− f)R ⊆⊕ RR, and so T = Ra ⊆⊕ RR.
(2). Observe that Re⊕Rf = Re⊕Rf(1−e), so Rf(1−e) ∼= Rf. Since R is left I-P -injective,
by (1), Rf(1− e) = Rh for some idempotent element h ∈ I. Let g = e+ h− eh. Then g2 = g such
that ge = g = eg and gh = h = hg. It follows that Re⊕Rf = Re⊕Rh = Rg.
(3). If Rk is a simple left ideal in I, and 0 6= ka ∈ kR, define γ : Rk → Rka; rk 7→ rka.
Then γ is an isomorphism, and so, as R is left I-P -injective, γ−1 = ·c for some c ∈ R. Then
k = γ−1(ka) = kac ∈ kaR. Therefore, kR is a simple right ideal.
(4). It follows from (3).
Proposition 6.1 is proved.
A ring R is called left Kasch if every simple left R-module embeds in RR, or equivalently,
rR(T ) 6= 0 for every maximal left ideal T of R. Right Kasch, right P -injective rings have been
discussed in [19]. Next, we discuss left Kasch left I-P -injective rings.
Proposition 6.2. Let R be a left I-P -injective left Kasch ring. Then:
(1) Soc(IR) ⊆ess IR.
(2) rI(J) ⊆ess IR.
Proof. (1). If 0 6= a ∈ I, let lR(a) ⊆ T, where T is a maximal left ideal. Then rR(T ) ⊆
⊆ rRlR(a) = aR, and (1) follows because rR(T ) is simple by [19] (Theorem 3.31).
(2). If 0 6= b ∈ I. Choose M maximal in Rb, let σ : Rb/M → RR be monic, and define γ : Rb→
→ RR by γ(x) = σ(x+M). Then γ = ·c for some c ∈ R by hypothesis. Hence bc = σ(b+M) 6= 0
because b /∈ M and σ is monic. But Jbc = γ(Jb) = 0 because Jb ⊆ M (if Jb * M, then
Jb+M = Rb. But Jb << Rb, so M = Rb, a contradition). So 0 6= bc ∈ bR ∩ rI(J), as required.
Proposition 6.2 is proved.
Recall that a left R-module M is called mininjective [17] if every R-homomorphism from a
minimal left ideal to M extends to a homomorphism of R to M.
Proposition 6.3. If M is a JP -injective left R-module, then it is mininjective.
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I-n-COHERENT RINGS, I-n-SEMIHEREDITARY RINGS AND I-REGULAR RINGS 785
Proof. Let Ra be a minimal left ideal of R. If (Ra)2 6= 0, then exists k ∈ Ra such that
Rak 6= 0. Since Ra is minimal, Rak = Ra. Thus k = ek for some 0 6= e ∈ Ra, this shows that
e2− e ∈ lRa(k). But lRa(k) 6= Ra because ek 6= 0, and note that Ra is simple, we have lRa(k) = 0,
and so e2 = e and Ra = Re. Clearly, in this case, every homomorphism from Ra to M can be
extended to a homomorphism of R to M. If (Ra)2 = 0, then a ∈ J(R). Since M is JP -injective,
every homomorphism from Ra to M can be extended to R.
Proposition 6.3 is proved.
Theorem 6.2. The following statements are equivalent for a ring R:
(1) R is an I-regular ring.
(2) Every left R-module is I-F -injective.
(3) Every left R-module is I-P -injective.
(4) Every cyclic left R-module is I-P -injective.
(5) Every left R-module is I-flat.
(6) Every left R-module is I-P -flat.
(7) Every cyclic left R-module is I-P -flat.
(8) R is left I-semihereditary and left I-F -injective.
(9) R is left IPP and left I-P -injective.
Proof. (2)⇒ (3) ⇒ (4); (5)⇒ (6) ⇒ (7); and (8)⇒ (9) are obvious.
(1)⇒ (2), (5), (8). Assume (1). Then it is easy to prove by induction that every finitely generated
left ideal in I is a direct summand of RR, so (2), (5), (8) hold.
(4) ⇒ (1). Let a ∈ I. Then by (4), Ra is I-P -injective, so that Ra is a direct summand of RR.
And thus (1) follows.
(7) ⇒ (1). Let a ∈ I. Then by (5), R/Ra is I-P -flat. This follows that Ra is I-P -pure in R
by Proposition 3.1(1). By Theorem 3.3(3), we have Ra
⋂
aR = aRa, and hence a = aba for some
b ∈ R. Therefore, R is an I-regular ring.
(9) ⇒ (1). Let a ∈ I. Since R is left I-P -injective, rRlR(a) = aR by Theorem 6.1(3). Since R
is left IPP, Ra is projective, so lR(a) = Re for some e2 = e ∈ R. Thus, aR = rR(Re) = (1− e)R
is a direct summand of RR, and hence a is regular.
Theorem 6.2 is proved.
Corollary 6.2. The following statements are equivalent for a ring R:
(1) R is a semiprimitive ring.
(2) Every left R-module is J-F -injective.
(3) Every left R-module is J-P -injective.
(4) Every cyclic left R-module is J-P -injective.
(5) Every left R-module is J-flat.
(6) Every left R-module is J-P -flat.
(7) Every cyclic left R-module is J-P -flat.
(8) R is left J-semihereditary and left J-F -injective.
(9) R is left JPP and left J-P -injective.
Corollary 6.3. The following statements are equivalent for a ring R:
(1) R is a regular ring.
(2) Every left R-module is F -injective.
(3) Every left R-module is P -injective.
(4) Every cyclic left R-module is P -injective.
(5) Every left R-module is flat.
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(6) Every left R-module is P -flat.
(7) Every cyclic left R-module is P -flat.
(8) R is left semihereditary and left F -injective.
(9) R is left PP and left P -injective.
Theorem 6.3. The following statements are equivalent for a ring R:
(1) R is a regular ring.
(2) R is a left I-semiregular I-regular ring.
Proof. (1) ⇒ (2) is trivial.
(2) ⇒ (1). Let M be any left R-module. Since R is I-regular, by Theorem 6.2, M is I-P -
injective. But R is left I-semiregular, by Theorem 2.2, M is P -injective. Hence, R is a regular ring
by Corollary 6.3.
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Received 08.07.12
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