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Near Approximations in Modules

Davvaz Bijan, Setyawati Dian Winda, Soleha, Mukhlash Imam, Subiono

Foundations of Computing and Decision Sciences

2021

Vol. 46, No. 4

319--337

Rough set theory is a mathematical approach to imperfect knowledge. The near set approach leads to partitions of ensembles of sample objects with measurable information content and an approach to feature selection. In this paper, we apply the previous results of Bagirmaz [Appl. Algebra Engrg. Comm. Comput., 30(4) (2019) 285-29] and [Davvaz et al., Near approximations in rings. AAECC (2020). https://doi.org/10.1007/s00200-020-00421-3] to module theory. We introduce the notion of near approximations in a module over a ring, which is an extended notion of a rough approximations in a module presented in [B. Davvaz and M. Mahdavipour, Roughness in modules, Information Sciences, 176 (2006) 3658-3674]. Then we define the lower and upper near submodules and investigate their properties.

The intersection graph of annihilator submodules of a module

Pejman S. B., Payrovi Sh., Babaei S.

Opuscula Mathematica

2019

Vol. 39, no. 4

577--588

Let R be a commutative ring and M be a Noetherian R-module. The intersection graph of annihilator submodules of M, denoted by GA(M) is an undirected simple graph whose vertices are the classes of elements of [formula], for a, b ∈ R two distinct classes [a] and [b] are adjacent if and only if [formula]. In this paper, we study diameter and girth of GA(M) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that GA(M) is complete if and only if ZR{M) is an ideal of R. Also, we show that if M is a finitely generated R-module with [formula] and [formula] and GA(M) is a star graph, then r(AnnR(M)) is not a prime ideal of R and [formula].