On subsemimodules of semimodules

Let R be a semiring and M an R-semimodule. Our objective is to investigate properties of prime subsemimodules of semimodules. We characterize prime subsemimodules with prime ideals. Finally we give the prime avoidance theorem for semimodules.

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.

Let 𝑀 be an 𝑅-semimodule and 𝑁 non-zero subsemimodule of 𝑀. We say that 𝑁 is an essential subsemimodule of 𝑀, if 𝑁 ∩ 𝐾 ≠ (0) for every nonzero subsemimodule 𝐾 of 𝑀. In this paper we study some useful results on essential subsemimodules and singular semimodule of semiring.

International Journal of Algebra, 2010

2011

The concept of weakly prime ideals in rings was introduced by D. D. Anderson and Eric Smith in 2003[1]. Further, this concept in semirings and modules has been studied by many authers(see [2-5], 10). We extend this concept for subsemimodule theory of semimodules over semirings. Indeed we prove : 1) If M is an entire R-semimodule and N is a weakly prime (weakly primary) subsemimodule of M, then (N : M) is a weakly prime (weakly primary) ideal of R. 2) Let N be a k-weakly prime (k-weakly primary) subsemimodule of an R-semimodule M. Then N is either prime (primary) or (N : M)N = 0. Finally, we obtain some characterizations of weakly prime and weakly primary subsemimodules of semimodules over semirings.

2020

Let R be a commutative semiring with identity and M be a unitary Rsemimodule. Let φ : S(M) → S(M) ∪ {∅} be a function, where S(M) is the set of all subsemimodules of M . A proper subsemimodule N of M is called φ-prime subsemimodule, if r ∈ R and x ∈M with rx ∈ N \φ(N) implies that r ∈ (N :R M) or x ∈ N . So if we take φ(N) = ∅ (resp., φ(N) = {0}), a φ-prime subsemimodule is prime (resp., weakly prime). In this article we study the properties of several generalizations of prime subsemimodules.

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.

Acta Universitatis Carolinae Mathematica Et Physica, 2008

The paper continues the investigation of semimodules. Main emphasis is laid on minimal (i.e., every proper subsemimodule has just one element) and congruence-simple semimodules. 1. Auxiliary results (A) This paper is a continuation of [1] and we use the same notation. Let sM be a (left) semimodule. For arbitrary elements x,w e M, put ((x: W)S,M =)(X:W) = = {se S | sx = w}. Similarly, if x e M and _4 _: M, put (x: A)i = {seS\sxe A}. 1.1 Lemma, (i) (x : w\) + (x : w 2) = (x : w { + w 2). (ii) If w\ 7-w 2 then (x : W\) n (x : w 2) = 0. Proof Obvious. • 1.2 Lemma. Assume that 2w = w. Then: (i) (x : w) + (x : w) = (x : w). (ii) (x:w)n(y:w) = (x + y: w).

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.

2015

The concept of semiring was introduced by Vandiver [1] in 1934. A semiring is a nonempty set R together with two associative binary operations, addition and multiplication, which is called a semiring if (i) addition is a commutative operation, (ii) there exists 0 ∈ R such that x + 0 = x = 0 + x, x0 = 0 = 0x for each x ∈ R, and (iii) multiplication distributes over addition both from left and right. Denote the sets of all nonnegative and residue classes of integers modulo n, respectively, by Z+0 and Zn. The set Z + 0 is a semiring under usual addition andmultiplication of nonnegative integers, but it is not a ring. Concepts of commutative semiring, semiring with identity 1, can be defined on the similar lines as in rings. All semirings in this paper are assumed to be commutative with identity 1 ̸ = 0. Generalizing the notion of exact sequence of modules over rings, Abuhlail [2], Al-Thani [3], and Bhambri and Dubey [4] introduced different notions of exact sequence of semimodules over...