Let R be a commutative ring with identity and M be a unitary R-module. An R-module M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N=IM. It is shown that over a Noetherian domain R with dim(R)≤ 1, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of R. Moreover, we give a characterization of finitely generated multiplication modules.

In this paper, we discuss the periodicity problems in the , finitely generated algebraic structures and exhibit their natural sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the imposing edifice of group theory. One of the deepest and important results of the related theory of , finite groups is a complete classification of all periodic groups, that is, the finite groups with periodic properties.

Universal left congruences on semigroups were studied in “Y. Dandan, V. Gould, T. Quinn-Gregson and R. Zenab, Semigroups with finitely generated universal left congruence, Monat. Math. 190 (2019) 689−724”. We consider universal congruences on acts over monoids and extend the results from semigroups to acts. Among other things, for an S-act AS with zero over a monoid S, we prove that being finitely generated of the universal congruence ωA and being pseudofinite of AS coincide.

Several characterizations of a ring R is given for which any non-zero finitely generated module M is retractable in the sense that HomR(M,N) is non-zero whenever N is a non-zero submodule of M. Such rings are called finite retractable. It is shown that any ring being Morita equivalent to a commutative ring is finite retractable. Also, if the commutative ring is semi-Artinian then any non-zero module is retractable. The class of finite retractable rings is shown to be closed under Morita equivalence and finite direct products. Moreover, for a finite retractable ring R which is a right order in a ring Q, it is shown that Q is also finite retractable.

Let R be a commutative ring with identity and M be a unitary R-module. First, we study multiplication R-modules M where R is a one dimensional Noetherian ring or M is a finitely generated R-module...

Let R be a commutative ring and A(R) be the set of all ideals with non-zero annihilators. Assume that A (R) = A(R)⧹ f(0)g and F(R) denote the set of all  nitely generated ideals of R. In this paper, we introduce and investigate the  nitely generated annihilating-ideal graph of R, denoted by AGF (R). It is the (undirected) graph with vertices AF (R) = A (R) \ F(R) and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some basic properties of AGF (R). For instance, it is shown that if R is not a domain, then AGF (R) has ascending chain condition on vertices if and only if R is Noetherian. We characterize all rings for which AGF (R) is a  nite, complete, star or bipartite graph. Next, we study diameter and girth of AGF (R). It is proved that diam(AGF (R)) ⩽ diam(AG(R)) and gr(AGF (R)) = gr(AG(R)):

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Let M be a finitely generated module over a regular local ring (R, n), and let M = {Mi} be an n-stable filtration on M. As a consequence of a recent result by Rossi and Sharifan in [14], we prove that if the i-th Betti numbers of M and (grM({M}) coincide with each other, then for each j ³ i the j-th Betti numbers of them are the same and Syzi(M) is a Koszul module provided that Syzi(grM(M) is component wise linear.