I Semipotency and the Total of Modules

Algebra Colloquium, 2008

Let U be a submodule of a module M. We call U a strongly lifting submodule of M if whenever M/U=(A+U)/U ⊕ (B+U)/U, then M=P ⊕ Q such that P ≤ A, (A+U)/U=(P+U)/U and (B+U)/U=(Q+U)/U. This definition is a generalization of strongly lifting ideals defined by Nicholson and Zhou. In this paper, we investigate some properties of strongly lifting submodules and characterize U-semiregular and U-semiperfect modules by using strongly lifting submodules. Results are applied to characterize rings R satisfying that every (projective) left R-module M is τ (M)-semiperfect for some preradicals τ such as Rad , Z2 and δ.

Kyungpook Mathematical Journal

We say a module a semicommutative module if for any and any , implies . This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and be a p.p.-module, then is a semicommutative module iff is an Armendariz module. For any ring R, R is semicommutative iff A(R, ) is semicommutative. Let R be a reduced ring, it is shown that for number and , is semicommutative ring but is not.

In this paper we study the notion of preradical on some subcategories of the category of semimodules and homomorphisms of semimodules. Since some of the known preradicals on modules fail to satisfy the conditions of preradicals, if the category of modules was extended to semimodules, it is necessary to investigate some subcategories of semimodules, like the category of subtractive semimodules with homomorphisms and the category of subtractive semimodules with ҽҟ-regular homomorphisms.

Journal of Algebra, 2004

Let R be a ring and C a class of right R-modules closed under finite direct sums. If we suppose that C has a set of representatives, that is, a set V(C) ⊆ C such that every M ∈ C is isomorphic to a unique element [M] ∈ V(C), then we can view V(C) as a monoid, with the monoid operation

Colloquium Mathematicum, 2012

We investigate when the direct sum of semi-projective modules is semiprojective. It is proved that if R is a right Ore domain with right quotient division ring Q = R and X is a free right R-module then the right R-module Q ⊕ X is semi-projective if and only if there does not exist an R-epimorphism from X to Q.

2010

P. J. Allen introduced the notion of a Q-ideal and a construction process was presented by which one can build the quotient structure of a semiring modulo a Q-ideal. Here we introduce the notion of QM -subsemimodule N of a semimodule M over a semiring R and construct the factor semimodule M/N . It is shown that this notion inherits most of the essential properties of the factor modules over a ring. Mathematics subject classification: 16Y60.

Journal of Algebra, 1970

Modules over semirings have been studied by C&k&y and others. Csik;iny has given a general algebraic characterization of varieties which are essentially classes of modules over semirings. In this paper, we give another characterization which is based on GrHtzer's notion of independence of elements in universal algebra. In doing so, we have answered one of the questions raised by Grgtzer in . Our methods are so close to the methods of category theory that it seemed natural to extend it to a characterization theorem for categories of modules; one such characterization is given in Section 3.

Communications in Algebra, 2007

Proceedings of The Indian Academy of Sciences-mathematical Sciences, 2009

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.

2016

Abstract. We say a module MR a semicommutative module if for any m ∈ M and any a ∈ R, ma = 0 implies mRa = 0. This paper gives various properties of reduced, Ar-mendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and MR be a p.p.-module, then MR is a semicommutative module iff MR is an Armendariz module. For any ring R, R is semicommutative iff A(R,α) is semicommu-tative. Let R be a reduced ring, it is shown that for number n ≥ 4 and k = [n/2], T kn (R) is semicommutative ring but T k−1n (R) is not. 1.