On (co)pure Baer injective modules
For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every modu...
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Інститут прикладної математики і механіки НАН України
2021
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irk-123456789-1887082023-03-12T01:28:58Z On (co)pure Baer injective modules Hamid, M.F. For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as Q-copure submodule of a Q-copure Baer injective module. Certain types of rings are characterized using properties of Q-copure Baer injective modules. For example a ring R is Q-coregular if and only if every Q-copure Baer injective R-module is injective. 2021 Article On (co)pure Baer injective modules / M.F. Hamid // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 219–226. — Бібліогр.: 4 назв. — англ. 1726-3255 DOI:10.12958/adm1209 2020 MSC: 16D50. http://dspace.nbuv.gov.ua/handle/123456789/188708 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as Q-copure submodule of a Q-copure Baer injective module. Certain types of rings are characterized using properties of Q-copure Baer injective modules. For example a ring R is Q-coregular if and only if every Q-copure Baer injective R-module is injective. |
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Hamid, M.F. |
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Hamid, M.F. On (co)pure Baer injective modules Algebra and Discrete Mathematics |
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Hamid, M.F. |
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Hamid, M.F. |
| title |
On (co)pure Baer injective modules |
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On (co)pure Baer injective modules |
| title_full |
On (co)pure Baer injective modules |
| title_fullStr |
On (co)pure Baer injective modules |
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On (co)pure Baer injective modules |
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on (co)pure baer injective modules |
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Інститут прикладної математики і механіки НАН України |
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2021 |
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http://dspace.nbuv.gov.ua/handle/123456789/188708 |
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On (co)pure Baer injective modules / M.F. Hamid // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 219–226. — Бібліогр.: 4 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT hamidmf oncopurebaerinjectivemodules |
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2025-07-16T10:53:37Z |
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“adm-n2” — 2021/7/19 — 10:26 — page 219 — #55
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 2, pp. 219–226
DOI:10.12958/adm1209
On (co)pure Baer injective modules
M. F. Hamid
Communicated by R. Wisbauer
Abstract. For a given class of R-modules Q, a module
M is called Q-copure Baer injective if any map from a Q-copure
left ideal of R into M can be extended to a map from R into
M . Depending on the class Q, this concept is both a dualization
and a generalization of pure Baer injectivity. We show that every
module can be embedded as Q-copure submodule of a Q-copure
Baer injective module. Certain types of rings are characterized using
properties of Q-copure Baer injective modules. For example a ring R
is Q-coregular if and only if every Q-copure Baer injective R-module
is injective.
Introduction
Let Q be a non-empty class of left R-modules. An exact sequence
0 → A
f
→ B
g
→ C → 0 (1)
of left R-modules is called Q-copure if every module in Q is injective with
respect to the sequence. In this case, f is called a Q-copure monomorphism
and g a Q-copure epimorphism [4, p.322]. If we denote by PI the class of
pure injective modules then the PI-copure sequences are exactly the pure
exact ones, see [4, p.290]. So not only does this concept dualize purity but
generalizes it as well. We will need the following lemma later.
2020 MSC: 16D50.
Key words and phrases: Q-copure submodule, Q-copure Baer injective module,
pure Baer injective module..
“adm-n2” — 2021/7/19 — 10:26 — page 220 — #56
220 On (co)pure Baer injective modules
Lemma 1. [4, p.323] For a given class of modules Q, the following hold.
1) Any pushout of a Q-copure monomorphism is a Q-copure monomor-
phism.
2) If g ◦ f is a Q-copure monomorhphism, where f : A → B and
g : B → C, then so is f .
For details about Q-copure submodules the reader is referred to sec-
tion 38 of [4].
Thani [3] introduced pure Baer injective modules as those modules
which are injective with respect to all pure exact sequences with the ring
R as a middle term. Here we study Q-copure Baer injective modules for
some given non-empty class of left R-modules Q, i.e. modules injective
with respect to all Q-copure sequences with R as a middle term. Pure
Baer injective modules are, now, a special case of Q-copure Baer injectives
by choosing Q = PI.
Unless otherwise stated the ring R is always associative with identity,
all modules are left unital R-modules, and Q is a non-empty class of
modules. If there is no confusion or if the class Q is known we will drop
the letter Q and just say copure sequences and copure Baer injective
modules.
1. Copure Baer injective modules
Definition 1. An R-module M is called Q-copure Baer injective if any
homomorphism from a Q-copure left ideal of R into M has an extension
to a homomorphism from R into M .
We will often write copure Baer injective and mean Q-copure Baer
injective for some given class Q, just like when we say module and homo-
morphism (or map) and mean R-module and R-homomrphism (or R-map)
for some given ring R.
Examples. 1) Injective modules are Q-copure Baer injective for any
class Q.
2) All pure Baer injective (and therefore all pure injective) modules
are PI-copure Baer injective.
3) Putting the class Q = {Z}, we see that none of the proper ideals of
Z is {Z}-copure. Hence all Z-modules are {Z}-copure Baer injective but
of course not all of them are injective.
4) We know that all Z-modules are pure Baer injective, however, not
all of them are Q-copure Baer injective for all classes Q. For example, let
“adm-n2” — 2021/7/19 — 10:26 — page 221 — #57
M. F. Hamid 221
M be any injective Z-module, so that all ideals of Z are {M}-copure in Z.
Hence the {M}-copure Baer injective modules are precisely the injective
ones. Therefore Z, for instance, is not {M}-copure Baer injective.
5) Let the ring R be Z4 and Q = {Z4}. Since Z4 is quasi injective, the
sequence 0 → Z2 → Z4 is Q-copure exact. It is in fact the only nontrivial
one! So, both of Z4 and Z3 are Q-copure Baer injective, while Z2 is not.
To see this consider the following diagram:
Z2 Z4
Z2
1Z2
which cannot be completed because Z2 is not a direct summand of Z4.
6) If Q is the class of simple modules then the class of copure Baer
injective modules equals the class M of modules injective with respect
to all inclusions I → R with I an s-pure ideal of a commutative ring R,
see [1].
7) Any module Q is, of course, {Q}-copure Baer injective but may
not, in general, be pure Baer injective.
The following proposition is easy to verify.
Proposition 1. 1) The direct product (resp., direct sum) of a (finite)
family of modules is copure Baer injective if and only if each factor is
copure Baer injective.
2) An R-module M is copure Baer injective if and only if
Ext(R/I,M) = 0 for every copure left ideal I of R.
Proposition 2. The class of copure Baer injective modules is closed
under extensions.
Proof. Let 0 → A → B → C → 0 be an exact sequence with A and C
copure Baer injective. Exactness of the sequence 0 → Ext(R/I,A) →
Ext(R/I,B) → Ext(R/I,C) → 0 gives, by Proposition 1, that Ext(R/I,
B) = 0 for any copure left ideal I of R.
Thani [3] introduced left pure hereditary rings as those rings whose
every pure left ideal is projective. Here, we define left copure hereditary
rings.
Definition 2. The ring R is called left Q-copure hereditary if every copure
left ideal of R is projective.
“adm-n2” — 2021/7/19 — 10:26 — page 222 — #58
222 On (co)pure Baer injective modules
Of course, left pure hereditary rings are PI-copure hereditary. We will
just say ‘left copure hereditary’ when the class Q is known.
Theorem 1. The following statements are equivalent:
1) The ring R is left copure hereditary.
2) The homomorphic image of any copure Baer injective R-module is
copure Baer injective.
3) The homomorphic image of any injective R-module is copure Baer
injective.
4) Any finite sum of injective R-modules is copure Baer injective.
Proof. (1) ⇒ (2) Consider the diagram
0 I R
M K 0
f
g
of R-modules, where I is a copure left ideal in R and M is a copure Baer
injective module. Projectivity of I gives the existence of a φ : I → M
such that gφ = f . Copure Baer injectivity of M gives a map φ′ : R → M
extending φ, hence gφ′ extends f and K is copure Baer injective. (2) ⇒ (3)
is trivial. (3) ⇒ (1) Let I be a copure left ideal of R and consider the
following diagram for a given R-module M :
0 I R
E(M) K 0
ι
f
g
where E(M) denotes the injective envelope of M . Since K is copure Baer
injective, there is a map h : R → K such that h|I = f . Projectivity of R
gives a σ : R → E(M) such that gσ = h, i.e. gσι = hι = f . This means
I is E(M)-projective, i.e. I is projective. (3) ⇒ (4) is clear. (4) ⇒ (3)
Similar to the proof of (4) ⇒ (3) in [3, Theorem 2.2].
2. Embedding in copure baer injective modules
The main result of this section is the following:
Theorem 2. Let Q be a non-empty class of R-modules. Every module can
be embedded as a Q-copure submodule in some Q-copure Baer injective
module.
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M. F. Hamid 223
We break the proof into the following three lemmas.
Lemma 2. Every module can be embedded in a copure Baer injective
module.
Proof. Given a module A, we want to show the existence of a copure Baer
injective module that contains A as a submodule. Consider the copure left
ideals I of R and the set F of all maps f : I → A. Thus, for any f ∈ F there
is a pushout B and a map g : R → B with g|I = f . The module B may
not be copure Baer injective, so put A0 = A, A1 = B and repeat the above
process with A replaced by A1 to give A2 and A0 ⊆ A1 ⊆ A2. Continuing
in this manner, we get a sequence A0 ⊆ · · · ⊆ An ⊆ An+1 ⊆ · · · , for all
n ∈ N. Put Aω =
⋃
An. Now, for each nonlimit ordinal repeat the above
process. If we get to a limit ordinal, say l, define Al =
⋃
{As, s < l}. Let t
be the smallest ordinal with cardinality bigger than that of the ring R,
i.e. |t| = |R|+ (the successor cardinal of |R|). For each s < t, we have
|s| < |t|, t is an initial ordinal and At =
⋃
{As, s < t}. Now, At is our
copure Baer injective module. To see this, let I be a copure left ideal
of R and f : I → At any map. For each r ∈ I, let s(r) be the smallest
ordinal such that f(r) ∈ As(r). Then s(r) < t and |s(r)| < |t| = |R|+.
Hence |s(r)| 6 |R|. Put p = sup{s(r), r ∈ R}. As each |s(r)| 6 |R|, we
must have |p| 6 |R| < |t|. Hence, p < t. Since t is a limit ordinal, we have
p + 1 < t. Therefore, for each r ∈ R, r ∈ As(r) ⊆ Ap ⊆ Ap+1 ⊆ At. So,
f(I) ⊆ Ap. Moreover, the map f : I → Ap can be extended to a map
g : R → Ap+1 with g|I = f . View g now as a map R → At. (The proof is
adapted from [2, p. 295].)
Of course, we know that every module can be embedded in an injective
(hence copure Baer injective) module. But this, unlike the next lemmas,
does not guarantee that the embedding is copure.
Lemma 3. Suppose that A0 ⊆ A1 ⊆ · · · is an ascending chain of modules
such that Ai is a copure submodule of Ai+1 for all i. Then, A0 is copure
in
⋃
Ai.
Proof. Let M be a member of the class Q and f0 : A0 → M a map which
extends, by assumption, to a map f1 : A1 → M , which in turn extends
to f2 : A2 → M , and so on. View the maps fi as sets of ordered pairs
(ai, f(ai)) with ai ∈ Ai for all i. Hence, it is clear that fi ⊆ fi+1 for all
i and if (x, y1), (x, y2) ∈ fi for some i then y1 = y2. Now, claim that
f = ∪fi is a (well-defined) homomorphism. To see this, let x ∈ ∪Ai, i.e.
x ∈ Ai for some i and (x, fi(x)) ∈ fi ⊆ f . If (x, y1), (x, y2) ∈ f , then
“adm-n2” — 2021/7/19 — 10:26 — page 224 — #60
224 On (co)pure Baer injective modules
(x, y1) ∈ fi and (x, y2) ∈ fj for some i and j. Without loss of generality,
assume i 6 j, so that (x, y1) and (x, y2) are both in fj and therefore
(x, y1) = (x, y2) and f is well-defined. To finish the proof, let x, y ∈ ∪Ai
so that x ∈ Ai and y ∈ ∪Aj for some i and j. Again assume i 6 j, so
fj(x) = fi(x). Now, for any r, s ∈ R, fj(rx + sy) = rfj(x) + sfj(y). So
f(rx+ sy) = rf(x) + sf(y).
Lemma 4. The embedding in Lemma 2 is copure.
Proof. The construction of Ai in Lemma 2 shows, by (1) of Lemma 1, that
Ai is copure in Ai+1 for all i, and by Lemma 3, A is copure in
⋃
An = Aω.
Again by Lemma 3, Aω is copure in Aω+1 and Aω+1 is copure in Aω+2
and so on. In other words, for every ordinal s < |R|+, we have either A
is copure in As if s is not a limit ordinal, or As =
⋃
u<sAu if s is a limit
ordinal. In either case, A is copure in At, as desired.
The embedding Theorem can be used in characterizing some copure
exact sequences.
Theorem 3. The sequence 0 → I
ι
→֒ R → R/I → 0 is copure exact if
and only if every copure Baer injective R-module is injective with respect
to it.
Proof. Necessity is clear. To prove sufficiency, let j : I → C be a copure
embedding in a copure Baer injective module C (Theorem 2). Therefore,
there exists a map f : R → C such that fι = j. But j is a copure
monomorphism, so by (2) of Lemma 1 ι is a copure monomorphism.
3. Characterization of rings using copure Baer injectivity
Thani [3] proved that for a left self injective ring R, the condition that
R/I is pure Baer injective for every essential left ideal I of R is enough
to make R/I pure Baer injective for all left ideals I of R. Using the same
line of argument, we generalize this to copure Baer injectivity.
Proposition 3. Let R be a left self injective ring. If R/J is copure Baer
injective for any essential left ideal J of R, then R/I is copure Baer
injective for any left ideal I of R.
Proof. Since R is injective, the injective envelope E(I) of I must be a
direct summand of R, for any left ideal I of R. Therefore, E(I) = Re for
some idempotent e ∈ E(I). Now for the map f : R → Re, defined by
“adm-n2” — 2021/7/19 — 10:26 — page 225 — #61
M. F. Hamid 225
1 7→ e, since I is essential in Re, f−1(I) must as well be essential in R
and, therefore by assumption, R/f−1(I) is copure Baer injective. Define
f : Re/I → R/f−1(I) by re+ I 7→ r+ f−1(I) and proceed as in the proof
of [3, Proposition 2.3].
By a left Q-copure split ring we mean a ring every Q-copure left ideal
of which is a direct summand (hence a principal ideal). Clearly, every left
pure split ring is left PI-copure split and if a ring R is left copure-split
then it is left copure hereditary. The Q-copure split rings are characterized
in the following Theorem.
Theorem 4. The following statements are equivalent:
1) The ring R is left copure split.
2) Every R-module is copure Baer injective.
3) Any copure left ideal of R is copure Baer injective.
4) (a) R is left copure hereditary, and
(b) Every free left R-module is copure Baer injective.
Proof. (1) ⇒ (2) Let M be an R-module. Since every left ideal I of R is
a direct summand, every map I → M into any R-module can easily be
extended to a map R → M . (2) ⇒ (3) is obvious. (3) ⇒ (1) Let I be a
copure left ideal of R. Copure Baer injectivity of I gives a homomorphism
R → I that extends the identity map of I, which means I is a direct
summand of R. (1) ⇒ (4)(a) and (2) ⇒ (4)(b) are immediate. (4) ⇒ (3)
Let I be a copure left ideal of R, hence projective by (a) and therefore a
direct summand of some free R-module F . From (b) it follows that F is
copure Baer injective and by Proposition 1, so is I.
Recall that a ring R is called left coregular if every left ideal of R is
copure in R [4, p.324].
Theorem 5. For a ring R the following statements are equivalent:
1) The ring R is left coregular.
2) Every copure Baer injective R-module is injective.
3) Every copure Baer injective R-module is quasi injective.
Proof. (1) ⇒ (2) ⇒ (3) are obvious. (2) ⇒ (1) By assumption, every
copure Baer injective R-module is injective with respect to any sequence
0 → I → R → R/I → 0, which must, therefore, be copure exact by
Theorem 3. (3) ⇒ (2) Let M be a copure Baer injective R-module.
Hence, by Proposition 1, so is M ⊕E(R) which must be quasi injective by
assumption. Therefore, M is injective with respect to E(R). In particular,
M is R-injective or injective by Baer condition.
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226 On (co)pure Baer injective modules
References
[1] Iuliu Crivei, s-Pure submodules, Int. J. Math. Math. Sci. 2005, 2005, pp.491–497.
[2] John Dauns, Modules and Rings, Cambridge University Press, 1994.
[3] Nada M. Al Thani, Pure Baer injective modules, Int. J. Math. Math. Sci. 20, 1997,
pp.529–538.
[4] Robert Wisbauer, Foundations of Module and Ring Theory. Gordon and Breach,
1991.
Contact information
Mohanad Farhan
Hamid
Department of Production Engineering and
Metallurgy, University of Technology-Iraq,
Baghdad, Iraq
E-Mail(s): 70261@uotechnology.edu.iq,
mohanadfhamid@yahoo.com
Received by the editors: 30.06.2018.
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