Form of filters of semisimple modules and direct sums
Some collections of submodules of a module defined by certain conditions are studied. A generalization of the notion of radical (preradical) filter is considered. We study the form of filters of semisimple modules and direct sums.
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irk-123456789-1523492019-06-11T01:25:20Z Form of filters of semisimple modules and direct sums Maturin, Yu. Some collections of submodules of a module defined by certain conditions are studied. A generalization of the notion of radical (preradical) filter is considered. We study the form of filters of semisimple modules and direct sums. 2013 Article Form of filters of semisimple modules and direct sums / Yu. Maturin // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 226–232. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC:16D90, 16D10. http://dspace.nbuv.gov.ua/handle/123456789/152349 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Some collections of submodules of a module defined by certain conditions are studied. A generalization of the notion of radical (preradical) filter is considered. We study the form of filters of semisimple modules and direct sums. |
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Maturin, Yu. Form of filters of semisimple modules and direct sums Algebra and Discrete Mathematics |
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Maturin, Yu. |
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Maturin, Yu. |
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Form of filters of semisimple modules and direct sums |
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Form of filters of semisimple modules and direct sums |
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Form of filters of semisimple modules and direct sums |
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Form of filters of semisimple modules and direct sums |
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Form of filters of semisimple modules and direct sums |
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form of filters of semisimple modules and direct sums |
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Інститут прикладної математики і механіки НАН України |
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Form of filters of semisimple modules and direct sums / Yu. Maturin // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 226–232. — Бібліогр.: 5 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT maturinyu formoffiltersofsemisimplemodulesanddirectsums |
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2025-07-13T02:52:57Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 2. pp. 226 – 232
c© Journal “Algebra and Discrete Mathematics”
Form of filters of semisimple modules
and direct sums
Yuriy Maturin
Communicated by A. I. Kashu
Abstract. Some collections of submodules of a module
defined by certain conditions are studied. A generalization of the
notion of radical (preradical) filter is considered. We study the form
of filters of semisimple modules and direct sums.
All rings are considered to be associative with unit 16=0 and all modules
are left and unitary.
Let R be a ring. Put
(N : f)M = {x ∈ M |f(x) ∈ N} ,
End(M)N = {f ∈ End(M)|f(M) ⊆ N} .
Let E be some non-empty collection of submodules of a left R-module
M . We consider the following conditions:
(1) L ∈ E,L ≤ N ≤ M ⇒ N ∈ E;
(2) L ∈ E, f ∈ End(M) ⇒ (L : f)M ∈ E;
(3) N,L ∈ E ⇒ N ∩ L ∈ E;
(4) N ∈ E,N ∈ Gen(M), L ≤ N ≤ M ∧
∧ ∀g ∈ End(M)N : (L : g)M ∈ E ⇒ L ∈ E;
(5) N,L ∈ E,N ∈ Gen(M) ⇒ N ∩ L ∈ E.
2010 MSC: 16D90, 16D10.
Key words and phrases: ring, module, filter.
Yu. Maturin 227
Consider a generalization of the notion of radical (preradical) filter
(see [3, 4]).
A non-empty collection E of submodules of a left R-module M satis-
fying ((1)), ((2)), ((3)) is called a preradical filter of M (see [4]).
A non-empty collection E of submodules of a left R-module M satis-
fying ((1)), ((2)), ((4)) is called a radical filter of M (see [4]). It is easy
to see that for every radical filter of M ((5)) is held.
A preradical (radical) filter E of a left R-module M is said to be
trivial if either E = {L|L ≤M} or E = {M}.
Let M be a semisimple left R-module with a unique homogeneous
component and let M = ⊕
i∈I
Mi, where Mi is simple for each i ∈ I.
If N = ⊕
i∈J
Ni, where Ni is simple for each i ∈ J and M ∼= N , then
Card(I) = Card(J).
Put
Cards(M) := Card(I).
Let M be a semisimple R-module with a unique homogeneous compo-
nent. If Cards(M) is infinite, then we set
Ep(M) := {L|L ≤M,Cards(M/L) < p},
where p is an infinite cardinal number.
Theorem 1. Let M be a semisimple R-module with a unique homoge-
neous component. If Cards(M) is infinite, then every non-trivial radical
[preradical] filter of M is of the form
Ep(M)
for some infinite cardinal number p ≤ Cards(M).
Proof. Let M be a semisimple R-module with a unique homogeneous
component, Cards(M) = ∞, and E a non-trivial radical [preradical] filter
of M . Put
q := Cards(M).
It is obvious that for each L ∈ E there exists H ≤ M such that
M = L⊕H. Hence CardsH ≤ q.
We claim that CardsH 6= q. Indeed, suppose, contrary to our claim,
that CardsH = q. Since M is a semisimple R-module with a unique ho-
mogeneous component, for some set I we have that M = ⊕
i∈I
Mi, where Mi
228Form of filters of semisimple modules and direct sums
is simple for each i ∈ I and for every i, j ∈ I there exists an isomorphism
fij : Mi → Mj . Hence CardI = Cards(M) = q. Taking into account that
Cards(M) is infinite, by (2.1) [5, p. 417],
q + q = q.
Consider a set X such that CardX = q and X∩I = ∅. Since q+q = q,
there exists a bijection w : X ∪ I → I. Put
Y := w(X), Z := w(I).
Therefore, I = Y ∪Z, Y ∩Z = ∅, q = CardI = CardY = CardZ. Now
we obtain M = A⊕B, where A = ⊕
i∈Y
Mi,B = ⊕
i∈Z
Mi. Since H ≤ M , there
exists an isomorphism u : H → ⊕
i∈T
Mi for some T ⊆ I (see Proposition
9.4 [1]). It is clear that CardsH = CardT = q. Whence q = CardY =
CardZ = CardT . Let g : Y → T, c : Z → T be bijections. Consider the
following maps:
G : A → H,C : B → H,
where
G(
∑
i∈Y
mi) = u−1(
∑
i∈Y
fi,g(i)(mi)),
(mi ∈ Mi(i ∈ I), Card{i ∈ Y |mi 6= 0} < ∞),
C(
∑
i∈Z
mi) = u−1(
∑
i∈Z
fi,c(i)(mi)),
(mi ∈ Mi(i ∈ I), Card{i ∈ Z|mi 6= 0} < ∞).
It is easily seen that these maps are isomorphisms. Let n, r : M → M
are maps such that n(a + b) = G(a), (a ∈ A, b ∈ B) and r(a + b) =
C(b), (a ∈ A, b ∈ B). It is clear that n, r : M → M are endomorphisms.
Since L ∩ H = 0 and G,C are isomorphisms, (L : n)M = B and (L :
r)M = A. As L ∈ E , by ((2)), we get B ∈ E and A ∈ E. By ((3)) or
((5)) , 0 = A ∩ B ∈ E. Consequently, E is trivial. This contradicts our
assumption. Hence CardsH < q. The natural isomorphism H ∼= M/L
implies that Cards(M/L) < q. Now we consider the set Ω of all cardinal
numbers v such that
v ≤ q∀L ∈ E : Cards(M/L) < v.
Yu. Maturin 229
Ω 6= ∅, because q ∈ Ω. By [2, p. 82] , there exists the least element
p belonging to Ω. Thus ∀L ∈ E : Cards(M/L) < p. It means that
E ⊆ Ep(M).
Let L ∈ Ep(M). Whence Cards(M/L) < p. We claim that there exists
D ∈ E such that Cards(M/L) ≤ Cards(M/D). Conversely, suppose that
∀D ∈ E : Cards(M/D) < Cards(M/L).
But Cards(M/L) < p ≤ q. Hence Cards(M/L) ∈ Ω. Since p
is the least element belonging to Ω, p ≤ Cards(M/L), contrary to
Cards(M/L) < p.
Now we have that there exists D ∈ E such that Cards(M/L) ≤
Cards(M/D). It is easily seen that for L,D there exist H,K ≤ M such
that M = L⊕H,M = D⊕K. Since M/L ∼= H,M/D ∼= K, Cards(H) ≤
Cards(K). Since H ≤ M and K ≤ M , there exist isomorphisms u : H →
⊕
i∈T
Mi for some T ⊆ I and w : K → ⊕
i∈S
Mi for some S ⊆ I. Therefore
CardT ≤ CardS. From this we have that there exists an injective map
γ : T → S.
Consider the following map:
ψ : ⊕
i∈T
Mi → ⊕
i∈S
Mi,
where
ψ(
∑
i∈T
mi) =
∑
i∈Y
fi,γ(i)(mi),
(mi ∈ Mi(i ∈ I), Card{i ∈ T |mi 6= 0} < ∞).
It is obvious that ψ is a monomorphism. Now consider the following map:
η : M → M,
where
η(l + h) = w−1ψu(h), (l ∈ L, h ∈ H).
It is clear that η ∈ End(M). Since D ∩ K = 0 and im η ⊆ K, for
every l ∈ K,h ∈ H: η(l + h) ∈ D ⇔ w−1ψu(h) ∈ D ⇔ w−1ψu(h) = 0.
Since u,w are isomorphisms and ψ is monomorphism, for every h ∈ H:
w−1ψu(h) = 0 ⇔ h = 0. From the above it follows that (D : η)M = L.
Since E is a radical [preradical] filter of M and D ∈ E, (D : η)M = L
shows that L ∈ E, by ((2)). It means that Ep(M) ⊆ E. But E ⊆ Ep(M).
Hence E = Ep(M)
230Form of filters of semisimple modules and direct sums
Theorem 2. If M is a left R-module such that M = M1 ⊕M2 ⊕ . . .⊕Mn,
where Mi = TrM (Mi) for each i ∈ {1, 2, . . . , n} and ∀S : S ≤ M ⇒ S ∈
Gen(M), then every radical [preradical] filter E of M is of the form
E = {J1 + J2 + . . .+ Jn|Ji ∈ Ei(i ∈ {1, 2, . . . , n})},
where Ei is a radical [preradical] filter of Mi for each i ∈ {1, 2, . . . , n}.
Proof. Let E be a radical [preradical] filter of M and M = M1 ⊕M2 ⊕
. . .⊕Mn, where Mi = TrM (Mi) for each i ∈ {1, 2, . . . , n}. Put
Ei := {fi(K)|K ∈E}
for each i ∈ {1, 2, . . . , n}, where fi : M → M,fi(m1 +m2 + . . .+mn) =
mi, (m1 ∈ M1,m2 ∈ M2, . . . ,mn ∈ Mn) for each i ∈ {1, 2, . . . , n}.
(1) Let L ∈ Ei, L ≤ N ≤ Mi. Hence there exists P ∈ E such that
L = fi(P ). Since L ≤ N , P ≤ f−1
i (N). By (1), f−1
i (N) ∈ E,
because P ∈ E. Therefore N = fi(f
−1
i (N)) ∈ Ei.
(2) Let L ∈ Ei, f ∈ End(Mi). Hence there exists P ∈ E such that
L = fi(P ). Consider
F : M → M,
where F : m1+m2+. . .+mi+. . .+mn 7→ f(mi), (m1 ∈ M1, . . . ,mn ∈
Mn). Thus F ∈ End(M).
We claim that fi((P : F )M ) ≤ (L : f)Mi
. Indeed, let xi ∈ fi((P :
F )M ). We have that xi ∈ Mi. Thus there exists x ∈ (P : F )M
such that fi(x) = xi. Hence f(xi) = F (x) ∈ P . It is clear that
f(xi) ∈ Mi. Therefore f(xi) = fi(f(xi)) ∈ fi(P ) = L. Whence
xi ∈ (L : f)Mi
. We obtain fi((P : F )M ) ≤ (L : f)Mi
.
Since P ∈ E and F ∈ End(M), (P : F )M ∈ E, by (2). (P : F )M ∈
E implies fi((P : F )M ) ∈ Ei. Since fi((P : F )M ) ≤ (L : f)Mi
, (1)
implies (L : f)Mi
∈ Ei.
(3) Let L,N ∈ Ei. Hence there exist P, T ∈ E such that L = fi(P )
and N = fi(T ). By (3) (for the preradical filter E), P ∩ T ∈ E.
Therefore fi(P ∩ T ) ∈ Ei. Since fi(P ∩ T ) ⊆ fi(P ) ∩ fi(T ) = L∩N
and fi(P ∩ T ) ∈ Ei, we obtain L ∩N ∈ Ei, by (1).
(4) Let N ∈ Ei, N ∈ Gen(Mi), L ≤ N ≤ Mi ∧ ∀g ∈ End(Mi)N : (L :
g)Mi
∈ Ei.
Hence N = fi(T ) for some T ∈ E. Since T ⊆ f−1
i (N), f−1
i (N) ∈ E,
by (1). And f−1
i (N) ∈ Gen(M). L ≤ N implies f−1
i (L) ≤ f−1
i (N).
Yu. Maturin 231
Let G be an arbitrary element of End(M)
f−1
i
(N). By Proposition
8.16 [1] , Ms = TrM (Ms) is a fully invariant submodule of M for
each s ∈ {1, 2, . . . , n}. Hence G(Ms) ⊆ Ms for each s ∈ {1, 2, . . . , n}.
Consider
g : Mi → Mi,m 7→ G(m), (m ∈ Mi).
Since ∀g ∈ End(Mi)N : (L : g)Mi
∈ Ei, there exists Yg ∈ Ei such
that g(Yg) ≤ L. Since G(Ms) ⊆ Ms for each s ∈ {1, 2, . . . , n},
G(f−1
i (Yg)) = G(M1 ⊕ . . .⊕ Mi−1 ⊕ Yg ⊕Mi+1 ⊕ . . .⊕Mn) ⊆
⊆ M1 ⊕ . . .⊕Mi−1⊕ G(Yg) ⊕Mi+1 ⊕ . . .⊕Mn =
= M1 ⊕ . . .⊕Mi−1⊕ g(Yg) ⊕Mi+1 ⊕ . . .⊕Mn ⊆
⊆ M1 ⊕ . . .⊕Mi−1⊕ L⊕Mi+1 ⊕ . . .⊕Mn = f−1
i (L).
Hence f−1
i (Yg) ⊆ (f−1
i (L) : G)M . Since Yg ∈ Ei, there exists
P ∈ E such that Yg = fi(P ). Thus P ⊆ f−1
i (Yg). Hence P ⊆
(f−1
i (L) : G)M &P ∈ E. By (1), (f−1
i (L) : G)M ∈ E. Since
f−1
i (N) ∈ E, f−1
i (N) ∈ Gen(M), f−1
i (L) ≤ f−1
i (N) ≤ M and
∀G ∈ End(M)
f−1
i
(N) : (f−1
i (L) : G)M ∈ E, obtain f−1
i (L) ∈ E.
Therefore L = fi(f
−1
i (L)) ∈ Ei.
Let J ∈ E. Put Ji := fi(J), (i ∈ {1, 2, . . . , n}). By Proposition 8.20 [1],
TrJ(M) = TrJ(M1 ⊕ M2 ⊕ . . . ⊕ Mn) =
n∑
i=1
TrJ(Mi). Since J ≤ M ,
TrJ (Mi) ≤ TrM (Mi) = Mi for any i ∈ {1, 2, . . . , n}, by Proposition 8.16 [1].
Hence TrJ(M) =
n
⊕
i=1
TrJ(Mi), because M = M1 ⊕M2 ⊕ . . .⊕Mn. Since
J ∈ Gen(M), TrJ(M) = J , by Proposition 8.12 [1]. Whence
J =
n
⊕
i=1
TrJ(Mi)&∀i ∈ {1, 2, . . . , n} : TrJ(Mi) ≤ Mi.
Therefore TrJ(Mi) = Ji for any i ∈ {1, 2, . . . , n}. Thus J = J1 + J2 +
. . .+ Jn, where J1 ∈ E1, J2 ∈ E2, . . . , Jn ∈ En.
Let Pi ∈ Ei for each i ∈ {1, 2, . . . , n}. Hence there exists Hi ∈ E such
that Pi = fi(Hi). Thus Hi ⊆ f−1
i (Pi). By (1), f−1
i (Pi) ∈ E. f−1
i (Pi) ∈
Gen(M) for any i ∈ {1, 2, . . . , n}. By ((3)) or ((5)), f−1
1 (P1) ∩ f−1
2 (P2) ∩
. . .∩f−1
n (Pn) ∈ E. Since f−1
i (Pi) = M1+. . .+Mi−1+Pi+Mi+1 +. . .+Mn
for any i ∈ {1, 2, . . . , n}, P1 + P2 + . . .+ Pn ∈ E.
232Form of filters of semisimple modules and direct sums
References
[1] F.W. Anderson, K.R. Fuller, Rings and categories of modules. Springer, Berlin-
Heidelberg-New York, 1973.
[2] F. Hausdorff, Set theory. Second edition. Chelsea Publishing Company, New York,
2005.
[3] A.I. Kashu, Radicals and torsions in modules. Stiintca, Chisinau, 1983. (in Russian).
[4] Yu.P. Maturin, Preradicals and submodules. Algebra and Discrete Mathematics,
Vol.10 (2010), No.1, pp. 88-96.
[5] W. Sierpinski, Cardinal and ordinal numbers. Second edition. PWN, Warsaw, 1965.
Contact information
Yuriy Maturin Institute of Physics, Mathematics and Computer
Science,Drohobych Ivan Franko State Pedagog-
ical University, Stryjska, 3, Drohobych 82100,
Lviv Region, Ukraine
E-Mail: yuriy_maturin@hotmail.com
URL: www.drohobych.net/ddpu/
Received by the editors: 16.10.2012
and in final form 14.08.2013.
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