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1.
The concept of right Rickart rings (or right p.p. rings) has been extensively studied in the literature. In this article, we study the notion of Rickart modules in the general module theoretic setting by utilizing the endomorphism ring of a module. We provide several characterizations of Rickart modules and study their properties. It is shown that the class of rings R for which every right R-module is Rickart is precisely that of semisimple artinian rings, while the class of rings R for which every free R-module is Rickart is precisely that of right hereditary rings. Connections between a Rickart module and its endomorphism ring are studied. A characterization of precisely when the endomorphism ring of a Rickart module will be a right Rickart ring is provided. We prove that a Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is precisely a Baer module. We show that a finitely generated module over a principal ideal domain (PID) is Rickart exactly if it is either semisimple or torsion-free. Examples which delineate the concepts and results are provided. 相似文献
2.
3.
Rachid Tribak 《代数通讯》2013,41(8):3190-3206
We introduce and study the notion of wd-Rickart modules (i.e. modules M such that for every nonzero endomorphism ? of M, the image of ? contains a nonzero direct summand of M). We show that the class of rings R for which every right R-module is wd-Rickart is exactly that of right semi-artinian right V-rings. We prove that a module M is dual Baer if and only if M is wd-Rickart and M has the strong summand sum property. Several structure results for some classes of wd-Rickart modules and dual Baer modules are provided. Some relevant counterexamples are indicated. 相似文献
4.
5.
By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End 𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map. 相似文献
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7.
An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M R . Several equivalent conditions to being strongly duo are given. If M R is strongly duo and reduced, then End R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M R is strongly duo if and only if either M ≈ R or M R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring. 相似文献
8.
ZhiXiangWU 《数学学报(英文版)》2005,21(2):249-260
In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results. 相似文献
9.
The well-known Schur's Lemma states that the endomorphism ring of a simple module is a division ring. But the converse is not true in general. In this paper we study modules whose endomorphism rings are division rings. We first reduce our consideration to the case of faithful modules with this property. Using the existence of such modules, we obtain results on a new notion which generalizes that of primitive rings. When R is a full or triangular matrix ring over a commutative ring, a structure theorem is proved for an R-module M such that End R (M) is a division ring. A number of examples are given to illustrate our results and to motivate further study on this topic. 相似文献
10.
Hongbo Zhang 《代数通讯》2013,41(4):1420-1427
An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A 1⊕ A 2 with M′? M, there are decompositions M′ = M 1⊕ M 2, B = B 1⊕ B 2, and A i = C i ⊕ D i (i = 1,2) such that M 1⊕ B 1 = C 1⊕ D 2 = M 1⊕ C 1 and M 2⊕ B 2 = D 1⊕ C 2 = M 2⊕ C 2. 相似文献
11.
Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). Let F be a fully invariant submodule of M and I?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings. 相似文献
12.
A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A1 ⊕ A2 with M′ ? M, there is a decomposition M′ =M1 ⊕ M2 such that A = M′ ⊕ [A1 ∩ (M1 ⊕ B)] ⊕ [A2 ∩ (M2 ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions. 相似文献
13.
A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a,b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x;σ]∕(xn+1), where R is a strongly regular ring with an endomorphism σ be an endomorphism of R. 相似文献
14.
We introduce the notions of “t-extending modules,” and “t-Baer modules,” which are generalizations of extending modules. The second notion is also a generalization of nonsingular Baer modules. We show that a homomorphic image (hence a direct summand) of a t-extending module and a direct summand of a t-Baer module inherits the property. It is shown that a module M is t-extending if and only if M is t-Baer and t-cononsingular. The rings for which every free right module is t-extending are called right Σ-t-extending. The class of right Σ-t-extending rings properly contains the class of right Σ-extending rings. Among other equivalent conditions for such rings, it is shown that a ring R is right Σ-t-extending, if and only if, every right R-module is t-extending, if and only if, every right R-module is t-Baer, if and only if, every nonsingular right R-module is projective. Moreover, it is proved that for a ring R, every free right R-module is t-Baer if and only if Z 2(R R ) is a direct summand of R and every submodule of a direct product of nonsingular projective R-modules is projective. 相似文献
15.
In this article the authors give the relation between a finitely-generated torsionfree Dedekind module M over a domain R and prime submodules of the 𝒪(M)-module M and the ring 𝒪(M). They also prove that M is a finitely-generated torsionfree Dedekind module over a domain R if and only if every semi-maximal submodule of R-module M is invertible. 相似文献
16.
Let R be a ring with identity, M a right R-module and S = End
R
(M). In this note, we introduce S-semicommutative, S-Baer, S-q.-Baer and S-p.q.-Baer modules. We study the relations between these classes of modules. Also we prove if M is an S-semicommutative module, then M is an S-p.q.-Baer module if and only if M[x] is an S[x]-p.q.-Baer module, M is an S-Baer module if and only if M[x] is an S[x]-Baer module, M is an S-q.-Baer module if and only if M[x] is an S[x]-q.-Baer module. 相似文献
17.
Let R be a ring. A module MR is said to be GC2 if for any N≤ M with N? M, N is a direct summand of M. In this article, we give some characterizations and properties of GC2 modules and their endomorphism rings, and many results on C 2 modules and GC2 rings are generalized to GC2 modules. 相似文献
18.
M. A. Turmanov 《Journal of Mathematical Sciences》2006,137(6):5336-5345
Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented
as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that
the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently
used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its
endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness
in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules
over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module
E
(G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary
ring R with the identity element. Indeed, if
R
M is an arbitrary unitary left module and M
+ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M
+) under the canonical ring homomorphism R → E(M
+). Then it holds that if
E(M+)
N is a pure submodule in
E(M+)
M
+, then
R
N is a pure submodule in
R
M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free
groups of finite rank will be investigated.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238,
2004. 相似文献
19.
The aim of this paper is twofold. On the one hand, we show that the dual Goldie dimension codim(End(M
R
)) of the endomorphism ring End(M
R
) of a module M
R
can be used as a measure of the dimension of the module M
R
. On the other hand, we prove under suitable hypotheses the validity of the Krull–Schmidt Theorem for infinite direct sums of modules with homogeneous semilocal endomorphism rings. 相似文献
20.
An R-module M is called almost injective if M is a supplement submodule of every module which contains M. The module M is called F-almost injective if every factor module of M is almost injective. It is shown that a ring R is a right H-ring if and only if R is right perfect and every almost injective module is injective. We prove that a ring R is semisimple if and only if the R-module R R is F-almost injective. 相似文献