Direct sums of semi-projective modules

Iraqi Journal of Science, 2022

Let A, and N are a semiring ,and a left A- semimodule, respectively. In this work we will discuss two cases: The direct summand of π-projective semi module is π-projective, while the direct sum of two π-projective semimodules in general is not π-projective . The details of the proof will be given. We will give a condition under which the direct sum of two π-projective semi modules is π-projective, as well as we also set conditions under which π-projective semi modules are projective.

2010

In semimodules over semiring category, split exact sequences, projective semimodules and generators was studied by many authors recently. As an analogue to module theory, in this paper, we extend those works by studying some properties and giving new variants and new characterizations in semimodules category. Mathematics Subject Classification: 16Y60

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.

Mathematische Annalen, 1970

A left module M over an arbitrary ring is called an RD-module (or an RS-module) if every submodule N of M with Rad(M) ⊆ N is a direct summand of (a supplement in, respectively) M. In this paper, we investigate the various properties of RD-modules and RS-modules. We prove that M is an RD-module if and only if M = Rad(M) ⊕ X, where X is semisimple. We show that a finitely generated RS-module is semisimple. This gives us the characterization of semisimple rings in terms of RS-modules. We completely determine the structure of these modules over Dedekind domains.

Turkish Journal of Mathematics

A right R-module M is called semi-projective if, for any submodule N of M , every epimorphism π : M → N and every homomorphism α : M → N , there exists a homomorphism β : M → M such that πβ = α (see [11]). In this paper, we consider some generalizations of semi-projective module, that is quasi pseudo principally projective module. Some properties of this class of module are studied.

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.

Russian Mathematics

Let $R$ be an associative ring with identity element. The authors prove that a projective left $R$-module $P$, whose ring of endomorphisms is semilocal, is finitely generated if and only if the dual Goldie dimensions of $P$ and of the factor module $P/J(P)$ of $P$ by its Jacobson radical $J(P)$ are equal.

2009

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module M R we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As consequences of these results, new characterizations of V-rings, right Noetherian V-rings and strongly semiprime rings are obtained. As such, a hereditary left Noetherian ring R is a finite direct sum of simple Noetherian right V-rings if and only if all finitely generated right R-modules are semi-endosimple.

Journal of Algebra, 1979

We prove here, among other results, that if R is a commutative noetherian ring and proJective R[xx, ,..., x,]-modules of rank < Krull dim R are extended, then finitely generated projective R[x, ,..., x,]-modules are extended. We also give an example of a nonfinitely generated projective module over an integral domain which contains no unimodular elements. All the rings in this paper are associative and commutative with unit and the algebras are associative with unit. We present first some variants of Quillen's theorem [lo, Theorem 11. Our arguments in this part are based on a solution of Serre's problem, communicated to the author by Professor L.N. Vasergtein (Moscow) and on the proof of [lo, Theorem 11. If S is a multiplicative set in a ring R, M and R-module and m EM, we denote by m, the image of m in M, under the canonical homomorphism M + MS. As usual if m is a maximal ideal of R we use the notation