Preradicals and characteristic submodules: connections and operations
For an arbitrary module M∈R-Mod the relation between the lattice Lch(RM) of characteristic (fully invariant) submodules of M and big lattice R-pr of preradicals of R-Mod is studied. Some isomorphic images of Lch(RM) in R-pr are constructed. Using the product and coproduct in R-pr four operations...
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Інститут прикладної математики і механіки НАН України
2010
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| Назва видання: | Algebra and Discrete Mathematics |
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| Цитувати: | Preradicals and characteristic submodules: connections and operations / A.I. Kashu // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 59–75. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1546032019-06-16T01:32:30Z Preradicals and characteristic submodules: connections and operations Kashu, A.I. For an arbitrary module M∈R-Mod the relation between the lattice Lch(RM) of characteristic (fully invariant) submodules of M and big lattice R-pr of preradicals of R-Mod is studied. Some isomorphic images of Lch(RM) in R-pr are constructed. Using the product and coproduct in R-pr four operations in the lattice Lch(RM) are defined. Some properties of these operations are shown and their relations with the lattice operations in Lch(RM) are investigated. As application the case RM=RR is mentioned, when Lch(RR) is the lattice of two-sided ideals of ring R. 2010 Article Preradicals and characteristic submodules: connections and operations / A.I. Kashu // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 59–75. — Бібліогр.: 8 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/154603 of2000 Mathematics Subject Classification:16D90, 16S90, 06B23. en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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For an arbitrary module M∈R-Mod the relation between the lattice Lch(RM) of characteristic (fully invariant) submodules of M and big lattice R-pr of preradicals of R-Mod is studied. Some isomorphic images of Lch(RM) in R-pr are constructed. Using the product and coproduct in R-pr four operations in the lattice Lch(RM) are defined. Some properties of these operations are shown and their relations with the lattice operations in Lch(RM) are investigated. As application the case RM=RR is mentioned, when Lch(RR) is the lattice of two-sided ideals of ring R. |
| format |
Article |
| author |
Kashu, A.I. |
| spellingShingle |
Kashu, A.I. Preradicals and characteristic submodules: connections and operations Algebra and Discrete Mathematics |
| author_facet |
Kashu, A.I. |
| author_sort |
Kashu, A.I. |
| title |
Preradicals and characteristic submodules: connections and operations |
| title_short |
Preradicals and characteristic submodules: connections and operations |
| title_full |
Preradicals and characteristic submodules: connections and operations |
| title_fullStr |
Preradicals and characteristic submodules: connections and operations |
| title_full_unstemmed |
Preradicals and characteristic submodules: connections and operations |
| title_sort |
preradicals and characteristic submodules: connections and operations |
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Інститут прикладної математики і механіки НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/154603 |
| citation_txt |
Preradicals and characteristic submodules: connections and operations / A.I. Kashu // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 59–75. — Бібліогр.: 8 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT kashuai preradicalsandcharacteristicsubmodulesconnectionsandoperations |
| first_indexed |
2025-07-14T06:39:11Z |
| last_indexed |
2025-07-14T06:39:11Z |
| _version_ |
1837603384601870336 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 9 (2010). Number 2. pp. 59 – 75
c© Journal “Algebra and Discrete Mathematics”
Preradicals and characteristic submodules:
connections and operations
A. I. Kashu
Abstract. For an arbitrary module M ∈ R-Mod the relation
between the lattice L
ch(RM) of characteristic (fully invariant) sub-
modules of M and big lattice R-pr of preradicals of R-Mod is studied.
Some isomorphic images of Lch(RM) in R-pr are constructed. Using
the product and coproduct in R-pr four operations in the lattice
L
ch(RM) are defined. Some properties of these operations are shown
and their relations with the lattice operations in L
ch(RM) are in-
vestigated. As application the case RM = RR is mentioned, when
L
ch(RR) is the lattice of two-sided ideals of ring R.
Introduction
Let R be a ring with unity and R-Mod denote the category of unitary left
R-modules. We denote by R-pr the class of all preradicals of the category
R-Mod. The ordinary operations of meet and join of preradicals transform
R-pr into a big lattice, which was studied in a series of works (see, for
example, [1]-[4]).
For an arbitrary module RM ∈ R-Mod, in the lattice L(RM) of all
submodules of RM we distinguish the sublattice L
ch(RM) of characteristic
(fully invariant) submodules with the order relation „ ⊆ ” (inclusion) and
the lattice operations „ ∩ ” (intersection) and „ + ” (sum).
The aim of this work is to clarify connection between the lattice
L
ch(RM) of characteristic submodules of an arbitrary module RM and
the big lattice R-pr of preradicals of R-Mod, as well as the application of
2000 Mathematics Subject Classification: 16D90, 16S90, 06B23.
Key words and phrases: preradical, lattice, characteristic submodule, product
(coproduct) of preradicals.
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60 Preradicals and characteristic submodules
obtained results to introducing four operations in L
ch(RM). For that the
following mappings are used:
αM : L
ch(RM) −→ R-pr, N → αM
N ,
ωM : L
ch(RM) −→ R-pr, N → ωM
N ,
where αM
N and ωM
N are the preradicals of R-pr defined by the rules:
αM
N (RX) =
∑
f :M →X
f(N), ωM
N (RX) =
⋂
f :X →M
f−1(N),
for every module RX ∈ R-Mod (see: [1, 4, 5]).
The mappings αM and ωM define the bijections:
L
ch(RM)
α
M
−−→ A
M = {αM
N
∣
∣N ∈ L
ch(RM)},
L
ch(RM)
ω
M
−−→ Ω
M = {ωM
N
∣
∣N ∈ L
ch(RM)},
which can be transformed in the lattice isomorphisms. Moreover, the
equivalence relation ∼=M defined in R-pr by the rule
r ∼=M s ⇔ r(M) = s(M)
determines the factor-lattice R-pr
/
∼=M= I
M , which is isomorphic to
the lattice L
ch(RM) and consists of the equivalence classes of the form
IMN = [αM
N , ωM
N ], where N ∈ L
ch(RM) and [αM
N , ωM
N ] is the interval in R-pr
containing all preradicals between αM
N and ωM
N . So we have:
L
ch(RM) ∼= A
M ∼= Ω
M ∼= I
M
(
= R-pr
/
∼=M
) (
Proposition 2.3
)
.
It is proved that the join of preradicals in the lattice A
M coincides
with their join in R-pr, and the meet of preradicals in Ω
M coincides with
their meet in R-pr (Propositions 2.4, 2.5).
Using the relations between L
ch(RM) and R-pr (the mappings αM
and ωM), as well as the product and coproduct in R-pr, four operations
in L
ch(RM) are defined:
1) α-product: K · N = αM
K αM
N (M);
2) ω-product: K ⊙ N = ωM
K ωM
N (M);
3) α-coproduct: (N : K) = (αM
N : αM
K )(M);
4) ω-coproduct: (N g
: K) = (ωM
N : ωM
K )(M),
for every characteristic submodules K, N ∈ L
ch(RM).
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A. I. Kashu 61
Properties of these operations are studied and some relations between
them and lattice operations of Lch(RM) are shown. For example, it is
proved that α-product is left distributive with respect to sum, ω-product
is left distributive with respect to intersection, α-coproduct is right dis-
tributive with respect to sum, and ω-coproduct is right distributive with
respect to intersection (Propositions 3.3, 3.4, 4.3, 4.4).
The case RM = RR is studied, i.e. when L
ch(RR) is the lattice of
two-sided ideals of the ring R: the mappings αR and ωR are specified, as
well as the respective operations (two of them coincide with the ordinary
product and sum of ideals).
1. Preliminary notions and results
In this auxiliary section we remind some notions and results necessary for
the basic material.
Let R be an arbitrary ring with unity and R-Mod is the category
of unitary left R-modules. A preradical r of the category R-Mod is a
subfunctor of identity functor, i.e. r is a function which associates to every
module M ∈ R-Mod a submodule r(M) ⊆ M such that f (r(M)) ⊆ r(M ′)
for every R-morphism f : M → M ′. We denote by R-pr the class of all
preradicals of the category R-Mod. The order relation „ ≤ ” in R-pr is
defined as follows:
r ≤ s ⇔ r(M) ⊆ s(M)
for every M ∈ R-Mod.
The operations „ ∧ ” (meet) and „ ∨ ” (join) in R-pr are defined by the
rules:
(
∧
α∈A
rα
)
(M) =
⋂
α∈A
rα(M),
(
∨
α∈A
rα
)
(M) =
∑
α∈A
rα(M)
for every family {rα |α ∈ A} ⊆ R-pr and M ∈ R-Mod.
Then R-pr (∧ ,∨) has the ordinary properties of lattices with the
difference that R-pr is not necessarily a set, and so it is called a big lattice.
This lattice was studied from different points of view in a series of works,
for example in [1]-[4].
Besides the lattice operations in R-pr an important role is played by
the following two operations r · s and (r : s) (product and coproduct of
preradicals), which are defined by the rules:
(r · s)(M) = r
(
s(M)
)
, [(r : s)(M)]
/
r(M) = s
(
M
/
r(M)
)
,
for every r, s ∈ R-pr and M ∈ R-Mod. Some properties and applications
of these operations can be found in [1], [4], etc. In particular, is true
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62 Preradicals and characteristic submodules
Lemma 1.1 ([1], p.36; [4], Theorem 8). For every preradicals of R-pr the
following relations hold:
(a)
(
∧
α∈A
rα
)
· s =
∧
α∈A
(rα · s);
(b)
(
∨
α∈A
rα
)
· s =
∨
α∈A
(rα · s);
(c)
(
r :
(
∧
α∈A
sα
))
=
∧
α∈A
(r : sα);
(d)
(
r :
(
∨
α∈A
sα
))
=
∨
α∈A
(r : sα). �
Every preradical r ∈ R-pr defines the following two classes of modules:
R(r) = {M ∈ R-Mod
∣
∣r(M) = M} is the class of r-torsion modules,
P(r) = {M ∈ R-Mod
∣
∣r(M) = 0} is the class of r-torsionfree modules.
For some types of preradicals these classes restore the preradical r
([1]-[3]).
A preradical r ∈ R-pr is called:
- idempotent if r
(
r(M)
)
= r(M) for every M ∈ R-Mod;
- radical if r
(
M
/
r(M)
)
= 0 for every M ∈ R-Mod;
- hereditary if r(N) = N ∩ r(M) for every N ⊆ M ∈ R-Mod;
- cohereditary if r(M
/
N) = (r(M) +N)
/
N for every N ⊆ M ∈ R-
Mod.
Now we remind some standard methods of the construction of some
preradicals by a module M ∈ R-Mod or by an ideal I of the ring R.
For a fixed module M ∈ R-Mod we can define an idempotent preradical
rM by the rule:
rM(X) =
∑
f :M →X
Imf,
for every module X ∈ R-Mod (i.e. rM(X) is the trace of M in X). This
idempotent preradical is defined by the class of modules generated by
module M :
R(rM) = Gen(RM) =
= {X ∈ R-Mod
∣
∣ ∃ epi
�
∑
α∈A
Mα → X → 0,Mα
∼= M} ([1]-[3]).
Dually, the module M ∈ R-Mod defines a radical rM by the rule:
rM(X) =
⋂
f :X →M
Ker f,
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A. I. Kashu 63
for every module X ∈ R-Mod (i.e. rM(X) is the reject of M in X). It is
determined by the class of modules cogenerated by M :
P(rM) = Cog(RM) =
= {X ∈ R-Mod
∣
∣ ∃mono 0 → X →
∏
α∈A
Mα,Mα
∼= M}.
Further, if an ideal I of ring R is fixed, then some associated preradicals
are known, in particular:
- idempotent radical rI , determined by the class
R(rI) = {RX
∣
∣ IX = X};
- torsion (hereditary radical) rI such that
P(rI) = {RX
∣
∣ Ix = 0 ⇒ x = 0};
- pretorsion (hereditary preradical) r(I) such that
R (r(I)) = {RX
∣
∣ IX = 0}, i.e. r(I)(X) = {x ∈ X
∣
∣ Ix = 0};
- cohereditary radical r(I) with
P (r(I)) = {RX
∣
∣ IX = 0}, i.e. r(I)(X) = IX (see: [1, 3, 7]).
Let M be an arbitrary R-module and L(RM) be the lattice of its
submodules. A submodule N ∈ L(RM) is called characteristic (or fully
invariant) in M if f(N) ⊆ N for every R-endomorphism f : RM → RM .
This means that N is an R-S-subbimodule of bimodule RMS, where
S = End (RM). We denote by L
ch(RM) the set of all characteristic
submodules of RM(0, M ∈ L
ch(RM)). It is clear that the intersection and
the sum of characteristic submodules are submodules of the same type,
so L
ch(RM) (⊆, ∩, +) is a complete sublatice of L(RM).
The following well known fact shows the relation between the charac-
teristic submodules of RM and preradicals of R-pr (see: [1, 4], etc.).
Lemma 1.2. A submodule N ∈ L(RM) is characteristic in RM if and
only if N = r(M) for some preradical r ∈ R-pr. �
For a characteristic submodule N ∈ L
ch(RM) many preradicals r ∈ R-
pr with N = r(M) can exist. To describe all preradicals with this property
we use the preradicals αM
N and ωM
N , defined by the rules:
αM
N (X) =
∑
f :M →X
f(N), ωM
N (X) =
⋂
f :X →M
f−1(N)
for every X ∈ R-Mod (see: [4]-[6]; in [1] these preradicals are defined for
every N ∈ L(RM) and are denoted by t(N⊆M) and t(N⊆M), respectively).
For every N ∈ L
ch(RM) the relation αM
N ≤ ωM
N is true. Moreover, the
following fact is proved (see [1, 4], etc.).
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64 Preradicals and characteristic submodules
Lemma 1.3. Let RM be a fixed module and N ∈ L
ch(RM). A preradical
r ∈ R-pr has the property r(M) = N if and only if r belongs to the
interval IMN = [αM
N , ωM
N ] of R-pr. �
So αM
N is the least among the preradicals r of R-pr with r(M) = N
and ωM
N is the greatest among such preradicals.
We remark that the similar results as in Lemmas 1.2 and 1.3 for special
types of preradicals (pretorsions, torsions, etc.) were obtained in [1, 8].
2. The relation between the lattices L
ch(RM) and R-pr
We fix an arbitrary module M ∈ R-Mod and consider the lattice L
ch(RM)
of characteristic submodules of RM . Using the indicated above construc-
tions, we obtain the mappings:
αM : L
ch(RM) −→ R-pr, N → αM
N ,
ωM : L
ch(RM) −→ R-pr, N → ωM
N .
We denote the images of these mappings as follows:
A
M = Im (αM) = {αM
N
∣
∣N ∈ L
ch(RM)},
Ω
M = Im (ωM) = {ωM
N
∣
∣N ∈ L
ch(RM)}.
From the definitions of preradicals αM
N and ωM
N immediately follows
Lemma 2.1. The mappings αM and ωM are isotone, i.e. they preserve
the order relation:
N ⊆ K ⇒ αM
N ≤ αM
K , ωM
N ≤ ωM
K .�
We denote by 0 and 1 the trivial preradicals of R-pr, i.e. 0(X) = 0
and 1(X) = X, for every X ∈ R-Mod. From the definitions it follows that
if N = 0, then αM
N = αM
0 = 0 and ωM
N = ωM
0 = rM , where rM is the radical
defined by rM(X) =
⋂
f :X →M
Ker f (see Section 1).
In the other extreme case when N = M we have:
a) αM
M = rM , where rM is the idempotent preradical defined by
rM(X) =
∑
f :M →X
Imf (see Section 1);
b) ωM
M = 1.
So we obtain
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Lemma 2.2. For every module M ∈ R-Mod the following relations hold:
1) αM
0 = 0, αM
M = rM ;
2) ωM
0 = rM , ωM
M = 1;
3) A
M ⊆ [0, rM ] ⊆ R-pr;
4) Ω
M ⊆ [rM , 1] ⊆ R-pr. �
From the definitions it is clear that if N,K ∈ L
ch(RM) and N 6= K,
then αM
N 6= αM
K , therefore we have the bijection
L
ch(RM) −→ A
M , N → αM
N .
Since N ⊆ K if and only if αM
N ≤ αM
K , the set A
M (≤) can be
transformed in a lattice such that for elements αM
N , αM
K ∈ A
M the meet is
αM
N∩K and the join is αM
N+K . Hence the indicated bijection becomes the
lattice isomorphism: Lch(RM) ∼= A
M .
Similarly, the mapping ωM determined a bijection from L
ch(RM)
into Ω
M , and the set Ω
M can be transformed in a lattice such that for
ωM
N , ωM
K ∈ Ω
M the meet will be ωN∩K and the join will be ωN+K . So we
have the lattice isomorphism: Lch(RM) ∼= Ω
M .
From the foregoing it follows that there exists one more possibility
to obtain in R-pr a lattice isomorphic to L
ch(RM). For the fixed module
M ∈ R-Mod we define in R-pr the equivalence relation ∼=M as follows:
r ∼=M s ⇔ r(M) = s(M),
where r, s ∈ R-pr. Then the lattice R-pr is divided into equivalence classes,
which by Lemma 1.3 have the form of intervals IMN . We denote:
I
M = R-pr
/
∼=M=
{
I
M
N = [αM
N , ωM
N ]
∣
∣N ∈ L
ch(RM)
}
.
On this set the order relation is defined by the rule:
I
M
N ≤ I
M
K ⇔ αM
N ≤ αM
K ⇔ ωM
N ≤ ωM
K ⇔ N ⊆ K,
where N,K ∈ L
ch(RM). In particular, the least elements of IM is the
interval [0, rM ] of R-pr, and the greatest element is the interval [rM , 1]
(see Lemma 2.2).
By the definitions it follows that the set I
M (≤) can be transformed
into a lattice by the operations:
I
M
N ∧ I
M
K = I
M
N∩K , I
M
N ∨ I
M
K = I
M
N+K .
Thus the mapping N → IMN defines a bijection which becomes the lattice
isomorphism: Lch(RM) ∼= I
M .
Totalizing the previous considerations we have
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66 Preradicals and characteristic submodules
Proposition 2.3. For every module M ∈ R-Mod the following lattices
are isomorphic:
L
ch(RM), A
M ,ΩM , I
M = R-pr
/
∼=M .�
We remark that for elements of AM and Ω
M besides the lattice opera-
tions defined above, we have the operations „ ∧ ” and „ ∨ ” in the lattice
R-pr, the results of which not necessarily belong to A
M or Ω
M . Now we
will compare these operations between them.
For the lattice A
M we have
Proposition 2.4. Let M be an arbitrary R-module. For every character-
istic submodules N,K ∈ L
ch(RM) we have the relation:
αM
N+K = αM
N ∨ αM
K ,
i.e. the join in A
M coincides with the join in R-pr. Furthermore, αM
N∩K ≤
αM
N ∧ αM
K and αM
N ∧ αM
K ∈ IMN∩K.
Proof. For submodules N,K ∈ L
ch(RM) by definitions we have:
(αM
N ∨ αM
K ) (M) = αM
N (M) + αM
K (M) = N +K,
hence αM
N ∨ αM
K ∈ IMN+K = [αM
N+K , ωM
N+K ] and so αM
N ∨ αM
K ≥ αM
N+K .
On the other hand, since the mapping αM is isotone, we have αM
N ∨
αM
K ≤ αM
N+K and so we obtain αM
N ∨ αM
K = αM
N+K . Also by the fact
that αM is isotone it follows αM
N∩K ≤ αM
N ∧ αM
K . Since (αM
N ∧ αM
K ) (M) =
αM
N (M) ∩ αM
K (M) = N ∩ K, we obtain that αM
N ∧ αM
K ∈ IMN∩K .
Now we study the same question for the lattice Ω
M .
Proposition 2.5. For every characteristic submodules N,K ∈ L
ch(RM)
is true the relation:
ωM
N∩K = ωM
N ∧ ωM
K ,
i.e. the meet in Ω
M coincides with the meet in R-pr. Furthermore, ωM
N+K ≥
ωM
N ∨ ωM
K and ωM
N ∨ ωM
K ∈ IMN+K .
Proof. Since (ωM
N ∧ ωM
K ) (M) = ωM
N (M) ∩ ωM
K (M) = N ∩ K, we have
ωM
N ∧ ωM
K ∈ IMN∩K = [αM
N∩K , ω
M
N∩K], therefore ωM
N∩K ≥ ωM
N ∧ ωM
K . The
inverse inclusion follows from isotony of ωM , which implies also the last
statement of proposition.
Example. If RM is ch-simple, i.e. L
ch(RM) = {0, M}, then I
M =
{IM0 , IMM}, where IM0 = [0, rM ], IMM = [rM , 1], and R-pr = IM0 ∪ IMM ,
A
M = {0, rM}, Ω
M = {rM , 1}.
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3. Operations in L
ch(RM) defined by the product in R-pr
The relation between the lattices Lch(RM) and R-pr, indicated in Section 2,
will be utilized to define some operations in L
ch(RM) with the help of
product and coproduct in R-pr. In this section we consider the operations
in L
ch(RM) which are obtained by the product in R-pr.
Since we fix the module M ∈ R-mod, in the rest of this paper for
simplicity we will omit the index M in the notations αM
N , ωM
N , etc. As was
mentioned above (Section 1) the product in R-pr is defined by (r ·s)(M) =
r (s(M)) and among the properties we remind that r · s ≤ r ∧ s and are
true the relations:
(
∧
α∈A
rα
)
· s =
∧
α∈A
(rα · s),
(
∨
α∈A
rα
)
· s =
∨
α∈A
(rα · s) (Lemma 1.1).
Definition 1. For every characteristic submodules K, N ∈ L
ch(RM) we
define:
K ·N = αK αN(M) = αK(N),
i.e. K ·N =
∑
f :M →N
f(K). The submodule K ·N will be called α-product
of submodules K and N in L
ch(RM).
Definition 2. For every characteristic submodules K, N ∈ L
ch(RM) we
define:
K ⊙N = ωK ωN(M) = ωK(N),
i.e. K ⊙ N =
⋂
f :N→M
f−1(N). The submodule K ⊙ N will be called ω-
product of submodules K and N in L
ch(RM).
From the definitions it is obvious that K ·N and K ⊙N are charac-
teristic submodules in RM . For every K ∈ L
ch(RM) we have αK ≤ ωK,
therefore αK(N) ⊆ ωK(N), i.e. K ·N ⊆ K ⊙N . Since the mapping ωM is
isotone, from N ⊆ M it follows:
K ⊙N = ωK(N) ⊆ ωK(M) = K,
and by Definition 2 K ⊙N = ωK(N) ⊆ N . So we obtain:
K ·N ⊆ K ⊙N ⊆ K ∩N
for every submodules K, N ∈ L
ch(RM).
Now we consider some particular cases.
a) If K ∩N = 0 (for example, if K = 0 or N = 0), then
K ·N = K ⊙N = 0.
b) If K = M , then since αM = rM and ωM = 1 (Lemma 2.2) we have:
M ·N = αM(N) = rM(N) =
∑
f :M →N
f(M);
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68 Preradicals and characteristic submodules
M ⊙N = ωM(N) = 1(N) = N ,
for every N ∈ L
ch(RM).
c) If N = M , then:
K ·M = αK(M) = K,
K ⊙M = ωK(M) = K,
for every K ∈ L
ch(RM).
Totalizing these observations we have
Lemma 3.1. 1) For every submodules K, N ∈ L
ch(RM) the following
relations are true:
K ·N ⊆ K ⊙N ⊆ K ∩N ;
2) K ·N = K ⊙N = 0, if K = 0 or N = 0;
3) K ·M = K ⊙M = K for every K ∈ L
ch(RM);
4) M ·N = rM(N),M ⊙N = N for every N ∈ L
ch(RM). �
From Definitions 1 and 2 and since the mappings αM and ωM are
isotone (Lemma 2.1) we obtain
Lemma 3.2. The operations „ · ” and „ ⊙ ” of Definitions 1 and 2 are
monotone in both variables:
K1 ⊆ K2 ⇒ K1 ·N ⊆ K2 ·N, K1 ⊙N ⊆ K2 ⊙N ;
N1 ⊆ N2 ⇒ K ·N1 ⊆ K ·N2, K ⊙N1 ⊆ K ⊙N2. �
Remark. In the paper [5] the product K · N is used for the study of
prime modules and prime preradicals.
In continuation we will investigate the concordance of introduced
operations „ · ” and „ ⊙ ” in L
ch(RM) with the lattice operations „ ∩ ” and
„ + ” in this lattice.
For the operation „ · ” of Lch(RM) we have
Proposition 3.3. For every submodules K1, K2, N ∈ L
ch(RM) the fol-
lowing relation is true:
(K1 +K2) ·N = (K1 ·N) + (K2 ·N),
i.e. the α-product is left distributive with respect to the sum of characteristic
submodules.
Proof. By Proposition 2.4 αK1+K2
= αK1
∨ αK2
, therefore
(K1 +K2) ·N = αK1+K2
(N) = (αK1
∨ αK2
)(N) =
= αK1
(N) + αK2
(N) = (K1 ·N) + (K2 ·N).
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A similar result takes place for the operation „ ⊙ ” of Lch(RM).
Proposition 3.4. For every submodules K1, K2, N ∈ L
ch(RM) the fol-
lowing relation is true:
(K1 ∩K2)⊙N = (K1 ⊙N) ∩ (K2 ⊙N),
i.e. the ω-product is left distributive with respect to the intersection of
characteristic submodules.
Proof. From Proposition 2.5 it follows ωK1∩K2
= ωK1
∧ ωK2
, hence
(K1 ∩K2)⊙N = ωK1∩K2
(N) = (ωK1
∧ ωK2
)(N) =
= ωK1
(N) ∩ ωK2
(N) = (K1 ⊙N) ∩ (K2 ⊙N).
As to the other possible relations of such types, from Lemma 3.2 follows
Proposition 3.5. In the lattice L
ch(RM) the following inclusions are
true:
1) K · (N1 +N2) ⊇ (K ·N1) + (K ·N2);
2) K ⊙ (N1 +N2) ⊇ (K ⊙N1) + (K ⊙N2);
3) K · (N1 ∩N2) ⊆ (K ·N1) ∩ (K ·N2);
4) K ⊙ (N1 ∩N2) ⊆ (K ⊙N1) ∩ (K ⊙N2);
5) (K1 ∩K2) ·N ⊆ (K1 ·N) ∩ (K2 ·N);
6) (K1 +K2)⊙N ⊇ (K1 ⊙N) + (K2 ⊙N). �
4. Operations in L
ch(RM) defined by the coproduct in R-pr
By analogy with the previous case now we will use the coproduct in R-pr
to define two operations in L
ch(RM). As we mentioned in Section 1, the
coproduct (r : s) in R-pr is defined by [(r : s)(X)]
/
r(X) = s
(
X
/
r(X)
)
for every X ∈ R-Mod. It is known that (r : s) ≥ r + s and the following
relations hold:
(
r :
(
∧
α∈A
sα
))
=
∧
α∈A
(r :sα),
(
r :
(
∨
α∈A
sα
))
=
∨
α∈A
(r :sα) (Lemma 1.1).
As before we fix the module RM and consider the lattice L
ch(RM) of
characteristic submodules of RM .
Definition 3. For every submodules N, K ∈ L
ch(RM) we define:
(N : K) = (αN : αK)(M),
i.e. (N : K)
/
N = αK(M
/
N) =
∑
f :M →M/N
f(K), or
(N : K) = π−1
(
αK(M
/
N)
)
, where π : M → M
/
N is the natural
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70 Preradicals and characteristic submodules
epimorphism. The submodule (N : K) will be called α-coproduct of
submodules N and K in L
ch(RM).
Definition 4. For every submodules N, K ∈ L
ch(RM) we define:
(N g
: K) = (ωN : ωK) (M),
i.e. (N g
: K)
/
N = ωK(M
/
N) =
⋂
f :M/N→M
f−1(K), or
(N g
: K) = π−1
(
ωK(M
/
N)
)
, where π : M → M
/
N is the natural
epimorphism. The submodule (N g
: K) will be called ω-coproduct of
submodules N and K in L
ch(RM).
Obviously (N : K), (N g
: K) ∈ L
ch(RM) and since αK ≤ ωK we
have αK(M
/
N) ⊆ ωK(M
/
N), so (N : K) ⊆ (N g
: K). Moreover,
from Definition 3 it follows that if we distinguish among all R-morphism
f : M → M
/
N the natural epimorphism π : M → M
/
N , then we have:
αK(M
/
N) =
∑
f :M →M/N
f(K) ⊇ π(K) = (K +N)
/
N,
therefore (N g
: K) = π−1
(
αK(M
/
N)
)
⊇ K +N . So we have:
N +K ⊆ (N : K) ⊆ (N g
: K)
for every N, K ∈ L
ch(RM).
We consider the defined operations for some extremal cases.
a) If N +K = M (for example, N = M or K = M), then
(N : K) = (N g
: K) = M . So we have:
(M : K) = M, (M g
: K) = M, (N : M) = M, (N g
: M) = M for
every K, N ∈ L
ch(RM).
b) If N = 0, then
(0 : K) = π−1
(
αK(M
/
0)
)
= αK(M) = K;
(0 g
: K) = π−1
(
ωK(M
/
0)
)
= ωK(M) = K.
c) If K = 0, then since α0 = 0 and ω0 = rM (Lemma 2.2) we obtain:
(N : 0) = π−1
(
α0(M
/
N)
)
= π−1
(
0(M
/
N)
)
= π−1(0) = N ;
(N g
: 0) = π−1
(
ω0(M
/
N)
)
= π−1
(
rM(M
/
N)
)
, i.e.
(N g
: 0)
/
N = rM(M
/
N).
Unifying these remarks we have
Lemma 4.1. 1) For every N, K ∈ L
ch(RM) the following relations hold:
N +K ⊆ (N : K) ⊆ (N g
: K);
2) (N : K) = (N g
: K) = M , if N = M or K = M ;
3) (0 : K) = (0 g
: K) = K for every K ∈ L
ch(RM);
4) (N g
: 0)
/
N = rM(M
/
N) for every N ∈ L
ch(RM). �
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From Definitions 3 and 4 follows
Lemma 4.2. The operations „ :” and „g: ” in L
ch(RM) are monotone in
both variables:
N1 ⊆ N2 ⇒ (N1 : K) ⊆ (N2 : K), (N1
g
: K) ⊆ (N2
g
: K);
K1 ⊆ K2 ⇒ (N : K1) ⊆ (N : K2), (N g
: K1) ⊆ (N g
: K2). �
Remark. In the paper [6] the submodule (N g
: K) is used for the defi-
nition of coprime submodule and for the study of coprime preradicals.
Similarly to Propositions 3.3 and 3.4 for the α-coproduct and ω-
coproduct some properties of distributivity can be shown.
Proposition 4.3. For every submodules N, K1, K2 ∈ L
ch(RM) the fol-
lowing relation hold:
(N : (K1 +K2)) = (N : K1) + (N : K2),
i.e. the α-coproduct is right distributive with respect to the sum of charac-
teristic submodules.
Proof. By Proposition 2.4 we have αK1+K2
= αK1
∨ αK2
and from Lemma
1.1 it follows:
(
αN : (αK1
∨ αK2
)
)
= (αN : αK1
) ∨ (αN : αK2
).
Therefore:
(N : (K1 +K2)) = (αN : αK1+K2
)(M) =
(
αN : (αK1
∨ αK2
)
)
(M) =
= [(αN : αK1
) ∨ (αN : αK2
)](M) = (αN : αK1
)(M) + (αN : αK2
)(M) =
= (N : K1) + (N : K2).
Proposition 4.4. For every submodules N, K1, K2 ∈ L
ch(RM) the fol-
lowing relation holds:
(N g
: (K1 ∩K2)) = (N g
: K1) ∩ (N g
: K2),
i.e. the ω-coproduct is right distributive with respect to the intersection of
characteristic submodules.
Proof. Applying Proposition 2.5 we have ωK1∩K2
= ωK1
∧ ωK2
and by
Lemma 1.1
(
ωN : (ωK1
∧ ωK2
)
)
= (ωN : ωK1
) ∧ (ωN : ωK2
). Consequently
(N : (K1 ∩K2)) = (ωN : ωK1∩K2
)(M) = [ωN : (ωK1
∧ ωK2
)](M) =
= [(ωN : ωK1
) ∧ (ωN : ωK2
)](M) = (ωN : ωK1
)(M) ∩ (ωN : ωK2
)(M) =
= (N g
: K1) ∩ (N g
: K2).
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72 Preradicals and characteristic submodules
In the other possible cases we obtain only inclusions, which follows
from Lemma 4.2.
Proposition 4.5. In the lattice L
ch(RM) the following relations hold:
1) (N : (K1 ∩K2)) ⊆ (N : K1) ∩ (N : K2);
2) (N g
: (K1 +K2)) ⊇ (N g
: K1) + (N g
: K2);
3) ((N1 ∩N2) : K) ⊆ (N1 : K) ∩ (N2 : K);
4) ((N1 ∩N2) g
: K) ⊆ (N1
g
: K) ∩ (N2
g
: K);
5) ((N1 +N2) : K) ⊇ (N1 : K) + (N2 : K);
6) ((N1 +N2) g
: K) ⊇ (N1
g
: K) + (N2
g
: K). �
Remark. The Propositions 3.3, 3.4, 4.3, 4.4 are true for arbitrary inter-
sections
⋂
α∈A
Kα and sums
∑
α∈A
Kα of characteristic submodules.
We complete this section by some remarks on the arrangement (recip-
rocal position) of some preradicals in R-pr, related by the defined above
operations in L
ch(RM).
1) If N, K ∈ L
ch(RM) then we have αK αN ∈ R-pr, the submodule
K · N = αK αN(M) and corresponding preradical αK·N ∈ R-pr.
From definition αK·N ≤ αK αN and these preradicals belong to the
equivalence class IK·N . From the relations K ·N ⊆ K⊙N ⊆ N ∩K
if follows αK·N ≤ αK⊙N ≤ αN∩K and since αM is isotone we have
αN∩K ≤ αN ∧ αK .
2) Submodules N, K ∈ L
ch(RM) define the submodule K ⊙ N =
ωK ωN(M) and the preradical ωK⊙N ∈ R-pr. We have ωK ωN ,ωK⊙N ∈
IK⊙N , so ωK⊙N ≥ ωK ωN . From the same relations K · N ⊆ K ⊙
N ⊆ K ∩ N if follows ωK·N ≤ ωK⊙N ≤ ωK∩N = ωN ∧ ωK (by
Proposition 2.5).
3) Similarly, if N, K ∈ L
ch(RM) we have (N : K) = (αN : αK)(M)
and preradical α(N :K) ∈ R-pr. Since α(N :K), (αN : αK) ∈ I(N :K), we
obtain α(N :K) ≤ (αN : αK). Using the relations N +K ⊆ (N : K) ⊆
(N g
: K) and Proposition 2.4, we have αN ∨αK = αN+K ≤ α(N :K) ≤
α(N e
: K).
4) Finally, submodules N, K ∈ L
ch(RM) define the submodule (N g
: K) =
(ωN : ωK)(M) and preradical ω(N e
: K) ∈ R-pr. We have ω(N e
: K) ≥
(ωN : ωK) ∈ I(N e
: K). From the same relations N + K ⊆ (N :
K) ⊆ (N g
: K) if follows ωN+K ≤ ω(N :K) ≤ ω(N e
: K) and since ωM
is isotone we have ωN ∨ ωK ≤ ωN+K .
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5. The case RM =RR
Now we specify briefly the situation when RM =RR, i.e. when L
ch(RR) is
the lattice of two-sided ideals of the ring R. We show the relation between
L
ch(RR) and R-pr, as well as the operations introduced above by the
mappings αM and ωM , using the product and coproduct in R-pr.
For ideal I = 0 we have α0 = 0 and ω0 = rR, where rR(X) =
⋂
f :X→R
Ker f for every X ∈ R-Mod, i.e. rR is the radical cogenerated
by the module RR (i.e. P(rR) = Cog (RR)).
In the other extreme case when I = RR we have αR = rR = 1, since
RR is a generator of R-Mod:
R(rR) = Gen(RR) = R-Mod.
From the other hand, ωR = 1 and so ωR = αR, therefore in the lattice
I
R = R-pr
/
∼=R the least element is the interval [0, rR] and the greatest
element is the degenerated interval IR, consisting of one preradical: αR =
ωR = 1.
Every ideal I ∈ L
ch(RR) determines in the lattice I
R = R-pr
/
∼=R
the equivalence class II = [αI , ωI ]. We concretize these preradicals. By
definition αI(X) =
∑
f :R→X
f(I) for every X ∈ R-Mod. The isomorphism
HomR(RR, RX) ∼=R X show that every R-morphism f : RR → RX has
the form fx : RR → RX, where x ∈ X and fx(r) = r x for every r ∈ R,
so fx(I) = Ix. Thus we obtain:
αI(X) =
∑
f :R→X
f(I) =
∑
x∈X
Ix = IX.
In such way αI coincides with the cohereditary radical r(I), defined by the
class of modules
P(r(I)) = {X ∈ R-Mod
∣
∣ IX = 0} (see Section 1).
From the other hand, the preradical ωI by definition acts as follows:
ωI(X) =
⋂
f :X→R
f−1(I) = {x ∈ X
∣
∣ f(x) ∈ I∀ f : RX → RR}
for every X ∈ R-Mod.
Now we show what the defined above four operations represent in the
case of the lattice L
ch(RR).
a) The α-product in L
ch(RR).
If J, I ∈ L
ch(RR) then by definition J · I = αJ(I) =
∑
f(J) for all
R-morphisms f : RR → RI. We apply again the canonical isomorphism
RI ∼= HomR(R, I), representing every f : RR → RI in the form fi :
RR → RI, where i ∈ I and fi(r) = r i for every r ∈ R, so fi(J) = Ji.
Therefore
J · I =
∑
i∈ I
fi(J) =
∑
i∈ I
Ji = JI,
where JI is the ordinary product of ideals in R. So we have the following
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74 Preradicals and characteristic submodules
conclusion: α-product in L
ch(RR) coincides with the ordinary product of
ideals in R.
b) The ω-product in L
ch(RR).
By the definition of operation „ ⊙ ” for ideals J, I ∈ L
ch(RR) we have:
J ⊙ I = ωJ(RI) =
⋂
f : I→R
f−1(J) = {i ∈ I
∣
∣ f(i) ∈ J∀ f : RI → RR}.
By previous results it follows that JI = J · I ⊆ J ⊙ I ⊆ J ∩ I and from
Proposition 3.4 we have: (J1 ∩ J2) ⊙ I = (J1 ⊙ I) ∩ (J2 ⊙ I). Since the
mapping ωR is isotone we obtain the inclusions similar to relations of
Proposition 3.5.
c) The α-coproduct in L
ch(RR).
For ideals I, J ∈ L
ch(RR) by definition we have (I : J) = (αI :
αJ)(RR), i.e. (I : J)
/
I = αJ(R
/
I) =
∑
f(J) for all f : RR → R(R
/
I).
By the isomorphism HomR(R, R
/
I) we can represent every R-morphism
f : RR → R(R
/
I) in the form fx+I : RR → R(R
/
I), where x+I ∈ R
/
I
and fx+I(r) = r(x + I) for every r ∈ R. Since fx+I(J) = J(x + I), we
obtain:
(I : J)
/
I =
∑
x+I∈R/I
fx+I(J) =
∑
x+I∈R/I
J(x+ I) =
= J(R
/
I) = (JR+ I)
/
I = (J + I)
/
I,
therefore (I : J) = I + J . So the α-coproduct in L
ch(RR) coincides with
the sum of ideals of R.
d) The ω-coproduct in L
ch(RR).
If I, J ∈ L
ch(RR) then by definition we have:
(I g
: J) = (ωI : ωJ)(R) = π−1
(
⋂
f :R/I→R
f−1(J)
)
,
where f : R(R
/
I) → RR are R-morphism and π : RR → R(R
/
I) is
the natural epimorphism. In other form:
(I g
: J) = {r ∈ R | f(r + I) ∈ J∀ f : R(R
/
I) → RR}.
From general results we have I + J = (I : J) ⊆ (I g
: J) and
(I g
: (J1 ∩ J2)) = (I g
: J1) ∩ (I g
: J2).
So in the case RM = RR two operations coincide with product and
sum of ideals, having two new operations which can present interest for
further investigations.
References
[1] L. Bican, T. Kepka, P. Nemec, Rings, Modules and Preradicals. Marcel Dekker,
New York, 1982.
[2] B. Stenström, Rings of Quotients. Springer Verlag, Berlin, 1975.
[3] A.I. Kashu, Radicals and Torsions in Modules. Ştiinţa, Chişinău, 1983 (in Russian).
[4] F. Raggi, J.R. Montes, H. Rincon, R. Fernandes-Alonso, C. Signoret, The lattice
structure of preradicals. Commun. in Algebra, 30(3) (2002), pp. 1533-1544.
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[5] F. Raggi, J. Rios, H. Rincon, R. Fernandes-Alonso, C. Signoret, Prime and irreducible
preradicals. J. of Algebra and its Applications, v. 4, N. 4 (2005), pp. 451-466.
[6] F. Raggi, J.R. Montes, R. Wisbauer, Coprime preradicals and modules. J. of Pure
and Applied Algebra, 200(2005), pp. 51-89.
[7] A.I. Kashu, On some bijections between ideals, classes of modules and preradicals
of R-Mod. Bulet. A.Ş.R.M. Matematica, N 2(36), 2001, pp. 101-110.
[8] R. Marcov, The characteristic submodules determined by pretorsions, torsions and
jansian torsions. Bulet. A.Ş.R.M. Matematica, N 3(28), 1998, pp. 71-80.
Contact information
A.I. Kashu Institute of Mathematics and Computer Sci-
ence, Academy of Sciences of Moldova, 5
Academiei str. Chişinău, MD−2028 Moldova
E-Mail: kashuai@math.md
URL:
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A. I. Kashu
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