共查询到20条相似文献,搜索用时 31 毫秒
1.
Let R be a ring with identity and let M be a unital left R-module. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N ? K but L ? K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N ∩ IM = IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M. 相似文献
2.
A. KHAKSARI M. ERSHAD H. SHARIF 《数学学报(英文版)》2006,22(4):1189-1196
Strongly irreducible submodules of modules are defined as follows: A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given. 相似文献
3.
4.
M. Ghirati 《代数通讯》2013,41(2):564-580
The submodules with the property of the title (N ? M is strongly essential in M if ∏ I N is essential in ∏ I M for any index set I) are introduced and fully investigated. It is shown that for each submodule N of M there exists a subset T ? M such that N + T is strongly essential submodule of M and (N:T) = Ann(T), T ∩ N = 0. Basic properties of these objects and several examples are given and the counterparts of the related concepts to essential submodules are also introduced and studied. It is shown that each maximal left ideal of a left fully bounded ring is either a summand or strongly essential. Rings over which no module has a proper strongly essential submodule are characterized. It is also shown that the left Loewy rings are the only rings over which the essential submodules and strongly essential submodules of any left module coincide. Finally, a new characterization of left FBN rings is observed. 相似文献
5.
《Quaestiones Mathematicae》2013,36(3-4):219-234
Abstract For a unital module V over a commutative ring R, let C denote the collection of cyclic submodules. The ring ?R(V;C) = {f ε EndR V |f(C) ?C, ?C εR (V;C) has been the object of several recent studies in which the structure of ?R(V;C) is related to the triple (V, R,C). Here we introduce a new ring HR(V;C) containing ?(V;C) and investigate its structure in terms of the parameters (V, R, C). 相似文献
6.
We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z 2(M) ⊕ S(M) (where Z 2(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions. 相似文献
7.
Let R be a commutative ring with identity, let M be an R-module, and let K
1, . . . ,K
n
be submodules of M: We construct an algebraic object called the product of K
1, . . . ,K
n
: This structure is equipped with appropriate operations to get an R(M)-module. It is shown that the R(M)-module M
n
= M . . .M and the R-module M inherit some of the most important properties of each other. Thus, it is shown that M is a projective (flat) R-module if and only if M
n
is a projective (flat) R(M)-module. 相似文献
8.
《Quaestiones Mathematicae》2013,36(3-4):527-536
Abstract Let K, S, T be subsets of a near-ring R. Then K is (S, T)-distributive if: s(k 1 + k 2)t = sk 1 t + sk 2 t, for each k 1, k 2 ε K, s ε S, t ε T; and K is (S, T)-d.g. on X if K is (S, X)-distributive and T is contained in the additive subgroup generated by X. This paper considers υ-primitivity and the associated ?υ radicals under various such conditions, particularly where S, T, and K are powers of R. Natural examples which illustrate and delimit the theory are given. 相似文献
9.
Carl Faith 《代数通讯》2013,41(4):1807-1810
A right R-module M is (Goldie) finite dimensional (= f.d.) if M contains no infinite direct sums of submodules.M is quotient f.d. (= q.f.d.) if M/K is f.d. for all submodules K.A submodule I of M is subdirectly irreducible (= SDI) if V is the intersection of all submodules S α of M that properly contain I, then V ≠ I, equivalentlyM/I has simple essential socle V/I. A theorem of Shock [74] states that a q.f.d. right module M is Noether-ian iff every proper submodule of M is contained in a maximal submodule. Camillo [77], proved a companion theorem: M is q.f.d. iff every submodule A ≠ 0 contains a finitely generated (= f.g) submodule S such that A/S has no maximal submodules. Using these two results, and an idea of Camillo [75], we prove the theorem stated in the title. 相似文献
10.
11.
Majid M. Ali 《代数通讯》2013,41(12):4479-4501
All rings are commutative with identity and all modules are unital. Anderson proved that a submodule N of an R-module M is multiplication (resp. join principal) if and only if 0(+) N is a multiplication (resp. join principal) ideal or R(M). The idealization of M. In this article we develop more fully the tool of idealization of a module, particularly in the context of multiplication modules, generalizing Anderson's theorems and discussing the behavior under idealization of some ideals and some submodules associated with a module. 相似文献
12.
KARIM SAMEI 《Proceedings Mathematical Sciences》2011,121(2):121-132
An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of Spec(M)\mbox{\rm Spec}(M). Also a finitely generated module M is a Baer module if and only if Spec(M)\mbox{\rm Spec}(M) is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N,
Ann(x) \not í (N:M).\mbox{\rm Ann}(x) \not \subseteq (N:M). When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM. 相似文献
13.
Mohammad Reza Vedadi 《Mediterranean Journal of Mathematics》2012,9(1):143-151
A right R-module M is called co-Hopfian if injective endomorphisms of M
R
are surjective. It is shown that E(M
R
) is co-Hopfian if and only if M
R
does not contain an infinite direct sum
?i ? \mathbbNWi{{\oplus_{i \in \mathbb{N}}W_{i}}} of submodules such that each W
i+1 essentially embeds in W
i
. For many modules M
R
, including modules over a right FBN or right duo ring with Krull dimension, it is proved that E(M
R
) is co-Hopfian if and only if
(\mathbbN){(\mathbb{N})} ↪̸ M
R
for every non-zero X
R
. For a ring which has enough uniforms, the class of modules with co-Hopfian injective envelope is the same as the class of
modules with finite uniform dimension if and only if there are only finitely many isomorphism classes of indecomposable injective
modules. 相似文献
14.
Jitendra N. Manocha 《Journal of Pure and Applied Algebra》1975,6(3):253-258
Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie. 相似文献
15.
16.
Ivo Herzog 《Algebras and Representation Theory》2007,10(2):135-155
Given an R-T-bimodule
R
K
T
and R-S-bimodule
R
M
S
, we study how properties of
R
K
T
affect the K-double dual M** = Hom
T
[Hom
R
(M, K), K] considered as a right S-module. If
R
K is a cogenerator, then for every R-S-bimodule, the natural morphism Φ
M
: M → M** is a pure-monomorphism of right S-modules. If
R
K is the minimal (injective) cogenerator and K
T
is quasi-injective, then M
** is a pure-injective right S-module. If
R
K is the minimal (injective) cogenerator, and T = End
R
K it is shown that K
T
is quasi-injective if and only if the K-topology on R is linearly compact. If the
R
K-topology on R is of finite type, then the natural morphism Φ
R
: R → R** is the pure-injective envelope of R
R
as a right module over itself.
The author is partially supported by NSF Grant DMS-02-00698. 相似文献
17.
George Voutsadakis 《代数通讯》2013,41(8):3093-3112
Let ? = ?F, R, ρ? be a system language. Given a class of ?-systems K and an ?-algebraic system A = ?SEN,?N,F??, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K A of A-systems in K as the collection of all members of K of the form 𝔄 = ? SEN,?N,F?,R 𝔄 ?, for some set of relation systems R 𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K , on the one hand, and local properties of K A and global properties connecting K A and K A′, whenever there exists an ?-morphism ? F,α? : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K A is an algebraic closure system, for every ?-algebraic system A, provided that K is closed under subsystems and reduced products. 相似文献
18.
Relative copure injective and copure flat modules 总被引:1,自引:0,他引:1
Let R be a ring, n a fixed nonnegative integer and In (Fn) the class of all left (right) R-modules of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n-copure injective (resp., n-copure flat) if (resp., ) for any N∈In. It is shown that a left R-module M over any ring R is n-copure injective if and only if M is a kernel of an In-precover f:A→B of a left R-module B with A injective. For a left coherent ring R, it is proven that every right R-module has an Fn-preenvelope, and a finitely presented right R-module M is n-copure flat if and only if M is a cokernel of an Fn-preenvelope K→F of a right R-module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions. 相似文献
19.
A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules. 相似文献
20.
《Quaestiones Mathematicae》2013,36(2):173-182
Abstract A ring R is called pseudo-commutative if for each x,y ε R there exists an integer n = n(x, y) for which xy = nyx. We first show that a generalization of a commutativity condition of Chacron and Thierrin implies pseudo-commutativity in rings; we then study pseudo-commutativity and commutativity in rings with constraints of the form xy = σkiyixi, where the ki are integers. 相似文献