BULETINUL ACADEMIEI DE ŞTIINŢE
A REPUBLICII MOLDOVA. MATEMATICA
Number 2(69), 2012, Pages 59–73
ISSN 1024–7696
On partial inverse operations in the lattice
of submodules
A. I. Kashu
Abstract. In the present work two partial operations in the lattice of submodules
L(R M ) are defined and investigated. They are the inverse operations for ω-product
and α-coproduct studied in [6]. This is the continuation of the article [7], in which the
similar questions for the operations of α-product and ω-coproduct are investigated.
The partial inverse operation of left quotient N ·c K of N by K with respect
to ω-product is introduced and similarly the right quotient N :\ K of K by N with
respect to α-coproduct is defined, where N, K ∈ L(R M ). The criteria of existence
of such quotients are indicated, as well as the different forms of representation, the
main properties, the relations with lattice operations in L(R M ), the conditions of
cancellation and other related questions are elucidated.
Mathematics subject classification: 16D90, 16S90, 06B23.
Keywords and phrases: Ring, module, lattice, preradical, (co)product of preradical,
left (right) quotient of submodules.
1
Introduction and preliminaries
The purpose of this work is the investigation of two partial inverse operations in
the lattice of submodules L(R M ) of an arbitrary left R-module R M .
M
Using the preradicals of standard types αM
N and ωN , in the article [6] four operations in L(R M ) were introduced and studied: α-product, ω-product, α-coproduct
and ω-coproduct (see also [5]). For two of these operations (for α-product and
ω-coproduct) in the work [7] the inverse operations were introduced: left quotient
with respect to α-product and right quotient with respect to ω-coproduct. These
quotients exist for every pair of submodules of R M .
In the present article in a similar manner the inverse operations are defined
for the other two operations (for ω-product and α-coproduct). In contrast to the
preceding cases these operations are partial, since they exist only in some special
cases. We will show the criteria of existence of these quotients, their main properties,
relations with the lattice operations of L(R M ) and other results.
Let R-Mod be the category of unitary left R-modules, where R is an associative
ring with unity. For an arbitrary left R-module R M we denote by L(R M ) the
lattice of submodules of R M and by Lch (R M ) the lattice of characteristic (fully
invariant) submodules of R M (i.e. of submodules N ⊆ M with f (N ) ⊆ N for every
R-endomorphism f : R M → R M ).
c A. I. Kashu, 2012
59
60
A. I. KASHU
A preradical r of the category R-Mod is by definition a subfunctor of identity
functor of this category i.e. r(M ) ⊆ M for every M ∈ R-Mod and f (r(M )) ⊆ M ′
for every R-morphism f : R M → R M ′ . The family of all preradicals of R-Mod will
be denoted by R-pr, andVit is a ”big”
W lattice with respect to the operations ∧ and ∨,
where the preradicals
rα and
rα for every {rα | α ∈ A} ⊆ R-pr are defined
α∈A
by the rules:
V
α∈A
α∈A
T
rα (X) =
rα (X),
α∈A
W
α∈A
P
rα (X) =
rα (X),
α∈A
for every left R-module R X [1, 3, 4].
An important role in the theory of preradicals is played by the following two
operations in R-pr:
1) the product r · s of preradicals r, s ∈ R-pr
(r · s)(X) = r s(X) , X ∈ R-Mod;
2) the coproduct r : s of preradicals r, s ∈ R-pr
[(r : s)(X)] r(X) = s X r(X) , X ∈ R-Mod.
In the study of various questions on preradicals the following two types of preradicals turn out to be very useful. Every fixed pair N ⊆ M , where N ∈ L(R M ),
defines the preradicals αM
and ωNM in R-pr by the rules:
N
P
T
αM
f (N ),
ωNM (X) =
f −1 (N ),
N (X) =
f: M →X
f: X→M
and ωNM the standard preradicals,
for every X ∈ R-Mod [1, 4]. We will call αM
N
associated to the pair N ⊆ M .
Two special cases are important:
a) the idempotent preradical r M defined by the module R M :
P
r M (X) =
Im f – the trace of M in X (i.e. r M = αM
M );
f: M →X
b) the radical rM defined by R M :
T
rM (X) =
Ker f – the reject of M in X (i.e. rM = ω0M ).
f: X→M
From the definition of preradical it is obvious that for every r ∈ R-pr and every
M ∈ R-Mod we have r(M ) ∈ Lch (R M ). Moreover, the submodule N ⊆ M is characteristic in M if and only if there exists a preradical r ∈ R-pr such that N = r(M ).
An important property of standard preradicals is that for every N ∈ Lch (R M ) we
have
r(M ) = N ⇔ αM
≤ r ≤ ωNM ,
N
M
i.e. αM
N is the least preradical of R-pr with r(M ) = N , and ωN is the greatest
preradical with this property.
Using the standard preradicals four operations in L(R M ) can be defined by the
following rules [2, 5, 6]:
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
61
1) α-product of K, N ∈ L(R M ):
K · N = αM
K (N ) =
P
f (K);
f: M →N
2) ω-product of K, N ∈ L(R M ):
M
K ⊙ N = ωK
(N ) =
T
f −1 (K);
f: N →M
3) α-coproduct of N, K ∈ L(R M ):
(N : K) N = αM
N) =
K (M
P
f (K);
f : M → M/N
4) ω-coproduct of N, K ∈ L(R M ):
M
(N :g K) N = ωK
(M N ) =
T
f −1 (K).
f : M/N → M
Some properties of these operations are indicated in the works [2, 5, 6], as well as
their relations with the lattice operations of L(R M ). Moreover, in the article [7] the
inverse operations for α-product and ω-coproduct are defined and investigated. We
remind the basic definitions.
Let K, N ∈ L(R M ). The left quotient of N by K with respect to α-product
(denoted by N /. K) is the greatest among submodules L ∈ L(R M ) with the condition L · K ⊆ N . Using the properties of α-product the left quotient N /. K can be
represented in the form:
P
N /. K =
{Lα ∈ L(R M ) | Lα · K ⊆ N }.
α∈A
Dually, for K, N ∈ L(R M ) the right quotient of K by N with respect to ω-coproduct
is defined as the least submodule L ∈ L(R M ) with the property N :g L ⊇ K. It is
denoted by N :e K and can be represented as
T
N :e K =
{Lα ∈ L(R M ) | N :g Lα ⊇ K}.
α∈A
In the work [7] some ”arithmetic” of these operations is developed, indicating
the basic properties and relations (in particular, their form in the case R M = R R).
In this work our task is to try to develop the similar theory for the other two
operations: for ω-product and α-coproduct. These cases have some distinguishing
features, which show the necessity to study them separately.
2
The left quotient with respect to ω -product
By definition, for K, N ∈ L(R M ) the ω-product of K and N is the following
submodule of M :
T
M
K ⊙ N = ωK
(N ) =
f −1 (N ) (see Section 1).
f: N →M
Now we remind some properties of this operation [6].
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A. I. KASHU
Proposition 2.1. 1) K ⊙ N ⊆ K ∩ N
T
M
2) K⊙ M = ωK
(M ) =
(so 0 ⊙ N = 0 and K ⊙ 0 = 0).
f −1 (K) is the largest characteristic submodule of
f: M →M
M which is contained in K; therefore, if K ∈ Lch (R M ), then K ⊙ M = K.
T
M
3) M ⊙ N = ωM
(N ) =
f −1 (M ) = N for every N ∈ L(R M ).
f: N →M
4) The operation of ω-product is monotone in both variables.
5) (K ⊙ N ) ⊙ L ⊇ K :g (N :g L) for every K, N, L ∈ L(R M ).
T
T
Kα ⊙ N =
(Kα ⊙ N ) for every Kα , N ∈ L(R M ).
6)
α∈A
7) If
α∈A
M = R R and J, I ∈ L(R R), then
T
J ⊙ I = ωJR (R I) =
f −1 (J) = {i ∈ I | f (i) ∈ J ∀ f : R J → R R}.
R
f : r J → rR
By analogy with the preceding cases (see [7]) we introduce the following definition
of the inverse operation for ω-product.
Definition 2.1. Let N, K ∈ L(R M ). The left quotient of N by K with respect
to ω-product is defined as the least submodule
L ∈ L(R M ) with the property
L ⊙ K ⊇ N . We denote this submodule by N ·e K. It is defined by the conditions:
a) (N ·e K) ⊙ K ⊇ N ;
b) if L ⊙ K ⊇ N for some L ∈ L(R M ), then L ⊇ N ·e K.
Another form of this definition is the
Proposition 2.2. N ⊆ L ⊙ K
⇔ N ·e K ⊆ L.
Proof. (⇒) is the condition b) of Definition 2.1.
(⇐) Let N ·e K ⊆ L. Then using a) and the monotony of ω-product we have
N ⊆ (N ·e K) ⊙ K ⊆ L ⊙ K, so N ⊆ L ⊙ K.
The following
statement is the answer to the question on the existence of the left
quotient N ·e K for a pair of submodules N, K ∈ L(R M ).
Proposition 2.3. Let N, K ∈ L(R M ). The left quotient N ·e K of N by K with
respect to ω-product exists if and only if N ⊆ K. In this case it can be represented
in the form:
T
N ·e K =
{Lα ∈ L(R M ) | Lα ⊙ K ⊇ N }.
α∈A
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
63
Proof. (⇒) If there exists the left quotient N ·e K, then
N ⊆ (N ·e K) ⊙ K ⊆ (N ·e K) ∩ K Proposition 2.1, 1) ,
therefore N ⊆ N ·e K and N ⊆ K.
(⇐) Let K ⊇ N . Then the set of submodules
{Lα ∈ L(R M ) | Lα ⊙ K ⊇ N }
is not empty, since it contains M by Proposition 2.1, 3) we have M ⊙ K = K ⊇ N .
Therefore we can consider the submodule
T
{Lα ∈ L(R M ) | Lα ⊙ K ⊇ N },
α∈A
for which by Proposition 2.1, 6) we obtain:
T
T
Lα ⊙ K =
(Lα ⊙ K) ⊇ N.
α∈A
α∈A
T
Lα is the least submodule of R M
Moreover, by the construction it is clear that
α∈A
T
with the property L ⊙ K ⊇ N . Therefore we have
Lα = N ·e K.
α∈A
Taking into account this result we can say that the operation of left quotient
of
Definition 2.1 is a partial operation and considering the left quotient N ·e K we
will suppose in continuation that N ⊆ K.
Remark. In T
the case N ⊆ K from the definition of ω-product we have: L ⊙ K =
M
ωL (K) =
f −1 (L), therefore
f: K→M
L⊙K ⊇ N
⇔ f −1 (L) ⊇ N ∀ f : K → M
⇔ f (N ) ⊆ L ∀ f : K → M .
This remark help us to represent the left quotient N ·e K in other form.
Proposition 2.4. Let K, N ∈ L(R M ) and N ⊆ K. Then
P
N ·e K =
f (N ) = αK
N (M ) .
f: K→M
In particular, N ·e K is a characteristic submodule of M .
P
Proof. Denote L =
f (N ). From the preceding remark we have:
f: K→M
f (N ) ⊆ L ∀ f : K → M
⇔ N ⊆ f −1 (L) ∀ f : K → M
⇔ L ⊙ K ⊇ N.
It remains to verify that L is the least submodule of M with the property
′
′
L⊙
PK ⊇ N . Let L′ ⊙ K ⊇ N′ . Then f (N) ⊆ L for every f : K → M , therefore
f (N ) ⊆ L , i.e. L ⊆ L . So L = N ·e K. The last statement of proposition
f: K→M
is obvious, because r(M ) ∈ Lch (R M ) for every r ∈ R-pr.
The following statement shows the concordance of left quotient N ·e K with the
order relation (⊆) of L(R M ).
64
A. I. KASHU
Proposition
2.5.
1) (The monotony in the numerator).
N1 ·e K ⊆ N2 ·e K for every K ⊇ N2 .
If N1 ⊆ N2 , then
2) (The antimonotony in the denominator). If K1 ⊆ K2 , then N ·e K1 ⊇
N ·e K2 for every N ⊆ K1 .
Proof. 1) Let N1 ⊆ N2 and K ⊇ N2 . Then by the definition of left quotient we have (N2 ·e K) ⊙ K ⊇ N2 ⊇ N1 , therefore by Proposition 2.2 we obtain
N2 ·e K ⊇ N1 ·e K.
2) Let K1 ⊆ K2 and N ⊆ K1 . Then by definition (N ·e K1 ) ⊙ K1 ⊇ N and
(N ·e K2 ) ⊙ K2 ⊇ N . Now the condition K1 ⊆ K2 and the monotony of ω-product
imply
N ⊆ (N ·e K1 ) ⊙ K1 ⊆ (N ·e K1 ) ⊙ K2 ,
therefore N ·e K2 ⊆ N ·e K1 .
In continuation
we will consider some particular cases, showing the values of the
left quotient N ·e K in dependence on the values of its terms N and K.
Proposition 2.6. 1) If N = K then
P
N ·e K = N ·e N =
f (N ) = r N (M ) (the trace of N in M ).
f : N→M
2) If N = 0, then for every K ⊆ M we have:
P
N ·e K = 0 ·e K =
f (0) = 0.
f : K→M
3) If K = M , then for every N ⊆ M we have:
P
N ·e M =
f (N ) = αM
N (M ),
f : M →M
which is the least characteristic
submodule of M , containing N ; therefore, if
N ∈ Lch (R M ), then N ·e M = N .
4) If N = M , then
for
the
existence
of
N
·e K we must have K = M , so
P
M ·e M =
f (M ) = M .
f : M →M
5) If K = 0, then for the existence of N ·e K we must have N = 0,
so 0 ·e 0 = 0.
It is interesting to clarify the relations between the left quotient N ·e K and the
lattice operations of L(R M ) (the sum and intersection of submodules).
Proposition 2.7. 1) For every submodules N1 , N2 , K ∈ L(R M ) with N1 + N2 ⊆ K
the following relation holds:
(N1 + N2 ) ·e K = (N1 ·e K) + (N2 ·e K).
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
65
2) For every submodules N1 , N2 , K ∈ L(R M ) with N1 + N2 ⊆ K the following
relation holds:
(N1 ∩ N2 ) ·e K ⊆ (N1 ·e K) ∩ (N2 ·e K).
3) For every submodules N, K1 , K2 ∈ L(R M ) with N ⊆ K1 ∩ K2 the following
relation holds:
N ·e (K1 + K2 ) ⊆ (N ·e K1 ) ∩ (N ·e K2 ).
4) For every submodules N, K1 , K2 ∈ L(R M ) with N ⊆ K1 ∩ K2 the following
relation holds:
N ·e (K1 ∩ K2 ) ⊇ (N ·e K1 ) + (N ·e K2 ).
Proof. 1) The inclusion (⊇) follows from the monotony in the numerator of the left
quotient Proposition 2.5, 1) .
(⊆) Denote L = (N1 ·e K) + (N2 ·e K). From the monotony of ω-product and
the relations L ⊇ N1 ·e K and L ⊇ N2 ·e K we obtain:
L ⊙ K ⊇ (N1 ·e K) ⊙ K ⊇ N1 ,
L ⊙ K ⊇ (N2 ·e K) ⊙ K ⊇ N2 .
Therefore N1 + N2 ⊆ L ⊙ K and by Proposition 2.2 (N1 + N2 ) ·e K ⊆ L.
The statements 2), 3) and 4) follow from the monotony or antimonotony of the
left quotient (Proposition 2.5).
Remark. The relations of Proposition 2.7 can be generalized for the infinite case,
for example:
P
P
Nα ·e K =
Nα ·e K),
α∈A
α∈A
where Nα ⊆ K for every α ∈ A.
In the following two statements some new properties of left quotient N ·e K are
indicated.
Proposition 2.8. For every submodules N, L, K ∈ L(R M ) such that N ⊆ L ⊆ K
the following relation holds:
(N ·e K) ·e (L ·e K) ⊆ N ·e L.
Proof. By definition L ⊆ (N ·e K) ⊙ K. From the monotony of ω-product and
”semiassociativity” of Proposition 2.1, 5) we obtain:
N ⊆ (N ·e L) ⊙ L ⊆ (N ·e L) ⊙ [(L ·e K) ⊙ K] ⊆ [(N ·e L) ⊙ (L ·e K)] ⊙ K.
Therefore
N
K
⊆
(N
L)
⊙
(L
K)
and
so
(N
K)
(L
e
e
e
e
e
·
·
·
·
·
·e K) ⊆
N ·e L.
Proposition 2.9. For every submodules N, K, L ∈ L(R M ) such that K ⊆ L the
following relation holds:
(N ⊙ K) ·e L ⊆ N ⊙ (K ·e L).
66
A. I. KASHU
Proof. By definition K ⊆ (K ·e L) ⊙ L. Using the monotony of ω-product and
Proposition 2.1, 5) we have:
N ⊙ K ⊆ N ⊙ [(K ·e L) ⊙ L] ⊆ [N ⊙ (K ·e L)] ⊙ L.
Therefore (N ⊙ K) ·e L ⊆ N ⊙ [K ·e L].
Some properties of the left quotient N ·e K are true in the additional assumption
that the operation of ω-product in L(R M ) is associative, i.e. (K ⊙ L) ⊙ L =
K ⊙ (N ⊙ L) for every K, N, L ∈ L(R M ) see Proposition 2.1, 5) .
Proposition 2.10. Let R M be a left R-module with the property that the operation
of ω-product in L(R M ) is associative. Then:
1) for every submodules N, K, L ∈ L(R M ) such that N ⊆ L ⊙ K the following
relation holds:
(N ·e K) ·e L = N ·e (L ⊙ K);
2) for every submodules N, K, L ∈ L(R M ) such that N ⊆ L the following relation holds:
(N ⊙ K) ·e (L ⊙ K) ⊆ N ·e L.
Proof. 1) (⊇) By definition N ⊆ (N ·e K) ⊙ K and N ·e K ⊆ [(N ·e K) ·e L] ⊙ L.
Multiplying the last relation on the right by K, by the monotony and associativity
of ω-product we obtain:
N ⊆ (N ·e K) ⊙ K ⊆ {[(N ·e K) ·e L] ⊙ L} ⊙ K = [(N ·e K) ·e L] ⊙ (L ⊙ K).
Therefore N ·e (L ⊙ K) ⊆ (N ·e K) ·e L.
(⊆) By the definition of left quotient and Proposition 2.1, 5) we have:
N ⊆ [N ·e (L ⊙ K)] ⊙ (L ⊙ K) ⊆ {[N ·e (L ⊙ K)] ⊙ L} ⊙ K.
Therefore N ·e K ⊆ [N ·e (L ⊙ K)] ⊙ L and so (N ·e K) ·e L ⊆ N ·e (L ⊙ K).
2) By definition N ⊆ (N ·e L) ⊙ L. Applying the monotony and associativity
of ω-product we have:
N ⊙ K ⊆ [(N ·e L) ⊙ L] ⊙ K = (N ·e L) ⊙ (L ⊙ K).
Therefore (N ⊙ K) ·e (L ⊙ K) ⊆ N ·e L.
The following results deal with the property of cancellation for the left quotient
with respect to ω-product.
Remark. By Proposition 2.2 implication (⇒) from L ⊙ N ⊆ L ⊙ N it follows
(L ⊙ N ) ·e N ⊆ L
for every L, N ∈ L(R M ), where L ⊙ N ⊆ L
∩N
⊆ N.
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
67
Proposition 2.11. Let N, K ∈ L(R M ). The following conditions are equivalent:
1) (N ⊙ K) ·e K = N ;
2) N = L ·e K for some submodule L ∈ L(R M ).
Proof. 1) ⇒ 2) is obvious.
2) ⇒ 1). Let N = L ·e K, where L ∈ L(R M ). By definition (L ·e K) ⊙ K ⊇
L,
and using the monotony of left quotient in the numerator Proposition 2.5, 1) we
obtain
(N ⊙ K) ·e K = [(L ·e K) ⊙ K] ·e K ⊇ L ·e K = N ,
i.e. (N ⊙ K) ·e K ⊇ N . From the preceding remark we have the inverse inclusion,
therefore (N ⊙ K) ·e K = N .
Proposition 2.12. Let N, K ∈ L(R M ) and N ⊆ K. The following conditions are
equivalent:
1) (N ·e K) ⊙ K = N ;
2) N = L ⊙ K for some submodule L ∈ L(R M ).
Proof. 1) ⇒ 2) is obvious.
2) ⇒ 1). Let N = L ⊙ K, where
L
∈
L(
R M ). By definition (N ·e K) ⊙ K ⊇ N
and by the last remark (L ⊙ K) ·e K ⊆ L. Now using the monotony of ω-product
we obtain:
(N ·e K) ⊙ K = [(L ⊙ K) ·e K] ⊙ K ⊆ L ⊙ K = N ,
therefore (N ·e K) ⊙ K = N .
3
The right quotient with respect to α -coproduct
In this section the similar questions are discussed as in the preceding one for the
operation of α-coproduct. We remind that the α-coproduct in R M of submodules
N, K ∈ L(R M ) is defined by the rule:
P
N : K = πN−1 αM
N ) = {m ∈ M | m + N ∈
f (K)},
K (M
f : M → M/N
where πN : M → M
N is the natural morphism, i.e.
(N : K) N = αM
N ) (see Section 1).
K (M
Some properties of operation of α-coproduct are enumerated in the following
statement [6].
Proposition 3.1. 1) N : K ⊇ N + K for every N, K ∈ L(R M ), so M : K = M
and N : M = M for every N, K ∈ L(R M ).
2) If N = 0, then 0 : K is the least characteristic submodule of M containing K
for every K ∈ L(R M ); therefore if K ∈ Lch (R M ), then 0 : K = K.
68
A. I. KASHU
3) If K = 0, then N : 0 = N for every N ∈ L(R M ).
4) The operation of α-coproduct is monotone in both variables.
5) If R M is a projective module, then
(N : K) : L ⊆ N : (K : L)
for every submodules N, K, L ∈ L(R M ).
P
P
6) N :
Kα =
(N : Kα ) for every N, Kα ∈ L(R M ).
α∈A
7) If R M =
α∈A
R
R, then N : K = KR + N for every left ideals N, K ∈ L(R R).
Now we introduce the inverse operation for the α-coproduct which in some sense
is dual to the preceding notion (Definition 2.1).
Definition 3.1. Let K, N ∈ L(R M ). The right quotient of K by N with respect
to α-coproduct is defined as the greatest among submodules L ⊆ M with the property N : L ⊆ K. We denote this submodule by N :\ K. It is determined by the
conditions:
a) N : (N :\ K) ⊆ K;
b) if N : L ⊆ K, then L ⊆ N :\ K.
The following simple remark is useful in applications.
Proposition 3.2. For every submodules K, N, L ∈ L(R M ) we have:
N :L ⊆ K
⇔ L ⊆ N :\ K.
Proof. Implication (⇒) is the condition b) of Definition 3.1.
(⇐) If L ⊆ N :\ K, then from the monotony of α-coproduct and condition a)
we have:
N : L ⊆ N : (N :\ K) ⊆ K, i.e. N : L ⊆ K.
Now we study the question on the existence of right quotient with respect to
α-coproduct.
Proposition 3.3. Let K, N ∈ L(R M ). The right quotient N :\ K of K by N with
respect to α-coproduct exists if and only if N ⊆ K. In this case we have:
P
N :\ K =
{Lα ∈ L(R M ) | N : Lα ⊆ K}.
α∈A
Proof. (⇒) If the right quotient N :\ K exists, then by definition N :(N :\ K)
⊆ K and since N + (N :\ K) ⊆ N : (N :\ K) Proposition 3.1, 1) , we obtain
N :\ K ⊆ K and N ⊆ K.
(⇐) Let N ⊆ K. Then the set of submodules of M
{Lα ∈ L(R M ) | N : Lα ⊆ K}
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
69
is not empty, since
it contains the submodule 0 by Proposition 3.1, 3) we have
N : 0 = N ⊆ K . So we can consider the submodule
P
L=
{Lα ∈ L(R M ) | N : Lα ∈ K},
α∈A
for which by Proposition 3.1, 6) we have
P
P
N:
Lα =
(N : Lα ) ⊆ K.
α∈A
α∈A
Therefore N : L ⊆ K and by construction it is clear that L is the greatest submodule of M with this property, so by definition L = N :\ K.
In such way we obtain a partial operation of right quotient N :\ K, which is
considered in the case N ⊆ K, the condition which provides its existence. The right
quotient N :\ K in this case can be represented in the following form.
Proposition 3.4. Let N, K ∈ L(R M ) and N ⊆ K. Then:
T
M/N
N :\ K =
f −1 (K N ) = ωK/N
(M ) .
f : M → M/N
Therefore N :\ K is a characteristic submodule of M .
T
Proof. Let L =
f −1 (K N ). Then for every f : M → M/N we have
f : M → M/N
P
L ⊆ f −1 (K N ), i.e. f (L) ⊆ K N . Therefore
f (L) ⊆ K N and
f : M → M/N
P
−1
−1
M
f (L) ⊆ K, i.e. πN αL (M N ) ⊆ K and by the definition of
πN
f : M → M/N
α-coproduct N : L ⊆ K.
Moreover, L is the greatest
of M with thisPproperty. Indeed,
if
P submodule
−1
′
′
′
N : L ⊆ K, then πN
f (L ) ⊆ K, therefore
f (L ) ⊆ K N ,
f : M → M/N
f : M → M/N
i.e. f (L′ ) ⊆ K N forevery f : M → M/N
. This means
that L′ ⊆ f −1 (K N )
T
for every f : M → M N , so L′ ⊆
f −1 (K N ) = L. This proves that
f : M → M/N
N :\ K = L.
Remark. As was mentioned above N :\ K ⊆ K, since by Proposition 3.1, 1)
N + (N :\ K) ⊆ N : (N :\ K) ⊆ K.
Now we show the behaviour of the right quotient N :\ K relative to the partial
order (⊆) of the lattice L(R M ).
Proposition 3.5. 1) (The monotony in the numerator). If K1 ⊆ K2 , then
N :\ K1 ⊆ N :\ K2 for every N ⊆ K1 ∩ K2 .
2) (The antimonotony in the denominator). If N1 ⊆ N2 , then N1 :\ K ⊇ N2 :\ K
for every K ∈ L(R M ) with N1 + N2 ⊆ K.
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A. I. KASHU
Proof. 1) By definition N : (N :\ K1 ) ⊆ K1 ⊆ K2 , therefore N :\ K1 ⊆ N :\ K2 .
2) The monotony of α-coproduct and the definition of right quotient imply:
N1 : (N2 : K) ⊆ N2 : (N2 :\ K) ⊆ K ,
therefore N2 :\ K ⊆ N1 :\ K.
In continuation we consider the right quotient N :\ K in some particular cases.
Proposition 3.6. 1) If N = K, then
N :\ K = {m ∈ M | f (m) ∈ K N = 0 ∀ f : M → M / N } =
T
=
Ker f = rM/N (M ),
f : M → M/N
which is the greatest characteristic submodule of M contained in N .
2) If N = 0, then for every K ∈ L(R M ) we have:
T
M
0 :\ K = ωK
(M ) =
f −1 (K),
f: M →M
which is the greatest characteristic submodule of M contained in K; therefore
if K ∈ Lch (R M ), then 0 :\ K = K.
3) If K = M , then for every N ∈ L(R M ) we have:
N :\ M = {m ∈ M | f (m) ∈ K N = M N ∀ f : M → M / N } = M .
4) If K = 0, then for the existence of right quotient we must have N = 0 and
0 :\ 0 = 0.
5) If N = M , then for the existence of right quotient we must have K = M and
M :\ M = M .
The following statement has to do with the relation between the right quotient
and the lattice operations of L(R M ) (the sum and intersection of submodules).
Proposition 3.7. 1) For every submodules N, K1 , K2 ∈ L(R M ) with N ⊆ K1 ∩ K2
the following relation holds:
N :\ (K1 ∩ K2 ) = (N :\ K1 )
∩ (N :\ K2 ).
2) N :\ (K1 + K2 ) ⊇ (N :\ K1 ) + (N :\ K2 ), where N ⊆ K1 ∩ K2 .
3) (N1 ∩ N2 ) :\ K ⊇ (N1 :\ K) + (N2 :\ K), where N1 + N2 ⊆ K.
4) (N1 + N2 ) :\ K ⊆ (N1 :\ K)
∩ (N2 :\ K),
where N1 + N2 ⊆ K.
Proof. 1) (⊆) follows from Proposition 3.5, 1).
(⊇) Denote L = (N :\ K1 ) ∩ (N :\ K2 ). Then using the monotony of α-coproduct
and the relations L ⊆ N :\ K1 and L ⊆ N :\ K2 , we obtain
N : L ⊆ N : (N :\ K1 ) ⊆ K1 ,
N : L ⊆ N : (N :\ K2 ) ⊆ K2 .
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
71
Therefore N : L ⊆ K1 ∩ K2 and by the definition of right quotient we have
L ⊆ N :\ (K1 ∩ K2 ).
The statements 2), 3) and 4) follow from the monotony and antimonotony of
right quotient (Proposition 3.5).
Remark. The Proposition 3.7 can be generalized for any set of submodules, in
particular:
T
T
N :\
Kα =
(N :\ Kα ).
α∈A
α∈A
In continuation we indicate some properties of the right quotient N :\ K which are
true
in the cases when the ”semiassociativity” of α-coproduct see Proposition 3.1,
5) or the associativity of this operation is supposed.
Proposition 3.8. If R M is a projective module, L, N, K ∈ L(R M ) and N ⊆ K,
then
(L : N ) :\ (L : K) ⊇ N :\ K.
Proof. By definition K ⊇ N : (N :\ K). Using the monotony of α-coproduct and
Proposition 3.1, 5) we have:
L : K ⊇ L : [N : (N :\ K)] ⊇ (L : N ) : (N :\ K),
therefore by Proposition 3.2 (L : N ) :\ (L : K) ⊇ N :\ K.
Proposition 3.9. Let R M be a left R-module with the property that in L(R M ) the
operation of α-coproduct is associative. Then the following relations hold:
1) L :\ (N :\ K) = (N : L) :\ K, where N : L ⊆ K;
2) (L :\ N ) :\ (L :\ K) ⊇ N :\ K, where L ⊆ N ⊆ K;
3) L :\ (N : K) ⊇ (L :\ N ) : K, where L ⊆ N ∩ K.
Proof. 1) (⊆) By definition of right quotient we have:
N : (N :\ K) ⊆ K,
L : [L :\ (N : K)] ⊆ N :\ K.
Using the monotony of α-coproduct and the associativity
of this operation the
”semiassociativity” of Proposition 3.1, 5) is sufficient we obtain:
(N : L) : [L :\ (N :\ K)] ⊆ N : [L : (L :\ (N :\ K))] ⊆ N : (N : K) ⊆ K.
Therefore L :\ (N :\ K) ⊆ (N : L) :\ K.
(⊇) By the associativity of α-coproduct and by the definition of right quotient
we have:
N : {L : [(N : L) :\ K]} = (N : L) : [(N : L) :\ K] ⊆ K.
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A. I. KASHU
So by Proposition 3.2 L : [(N : L) :\ K] ⊆ N :\ K and (N : L) :\ K ⊆ L :\ (N :\ K).
2) By definition N ⊇ L : (L :\ N ). Using the monotony and associativity of
α-coproduct we obtain:
K ⊇ N : (N :\ K) ⊇ [L : (L :\ N )] : (N :\ K) = L : [(L :\ N ) : (N :\ K)].
By Proposition 3.2 it follows that L :\ K
(L :\ N ) :\ (L :\ K) ⊇ N :\ K.
⊇
(L :\ N ) : (N :\ K) and
3) By definition N ⊇ L : (L :\ N ), therefore by the monotony and associativity
of α-coproduct we have:
N : K ⊇ [L : (L :\ N )] : K = L : [(L :\ N ) : K]
and so L :\ (N : K) ⊇ (L :\ N ) : K.
The following subject of investigation is the cancellation property related to the
right quotient N :\ K. We begin with a simple
Remark. N :\ (N : L) ⊇ L for every N, L ∈ L(R M ), since denoting K = N : L
from the relation N : L ⊆ K we have L ⊆ N :\ K (Proposition 3.2).
Proposition 3.10. Let K, N ∈ L(R M ). The following conditions are equivalent:
1) N = K :\ (K : N );
2) N = K :\ L for some L ∈ L(R M ) with K ⊆ L.
Proof. 1) ⇒ 2) is obvious.
2) ⇒ 1). Let N = K :\ L, where L ∈ L(R M ) and K ⊆ L. By the definition of
right quotient and its monotony in the numerator we have:
K : (K :\ L) ⊆ L, K :\ [K : (K :\ L)] ⊆ K :\ L.
From the preceding remark in the last relation the inverse inclusion holds, therefore
N = K :\ L = K :\ [K : (K :\ L)] = K :\ (K : N ).
Proposition 3.11. Let K, N ∈ L(R M ) and K ⊆ N . The following conditions are
equivalent:
1) N = K : (K :\ N );
2) N = K : L for some submodule L ∈ L(R M ).
Proof. 1) ⇒ 2) is obvious.
2) ⇒ 1). Let N = K : L, where L ∈ L(R M ). By the preceding remark
K :\ (K : L) ⊇ L and using the monotony of α-coproduct we have:
K : [K :\ (K : L)] ⊇ K : L.
The inverse inclusion follows from the definition of right quotient, therefore:
N = K : L = K : [K :\ (K : L)] = K : (K :\ N ).
73
ON PARTIAL INVERSE OPERATIONS IN THE LATTICE . . .
Finally, we consider the particular case R M = R R. It is known that for
every
left ideals N, K ∈ L(R M ) we have N : K = KR + N Proposition 3.1, 7) .
Proposition 3.12. Let N, K ∈ L(R R) and N ⊆ K. Then
N : K = {r ∈ R | r·R ⊆ K},
the annihilator of R (R K) in R.
Proof. Denoting the right side of indicated relation by L, we have LR ⊆ K and
N : L = LR + N ⊆ K + N = K, i.e. N : L ⊆ K. Moreover, L is the greatest
submodule of M with this property. Indeed, if N : L′ ⊆ K, then N : L′ =
L′ R + N ⊆ K and L′ R ⊆ K, so by the definition of L it follows that L′ ⊆ L. Now
it is clear that N :\ K = L.
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A. I. Kashu
Institute of Mathematics and Computer Science
Academy of Sciences of Moldova
5 Academiei str. Chişinău, MD−2028
Moldova
E-mail: [email protected]
Received
May 15, 2012