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Direct Sums of Strongly Lifting Modules
Shahabaddin Ebrahimi Atani,Mehdi Khoramdel,Saboura Dolati Pishhesari 경북대학교 자연과학대학 수학과 2020 Kyungpook mathematical journal Vol.60 No.4
For the recently defined notion of strongly lifting modules, it has been shown that a direct sum is not, in general, strongly lifting. In this paper we investigate the question: When are the direct sums of strongly lifting modules, also strongly lifting? We introduce the notion of a relatively strongly projective module and use it to show if M = M1+M2 is amply supplemented, then M is strongly lifting if and only if M1 and M2 are relatively strongly projective and strongly lifting. Also, we consider when an arbitrary direct sum of hollow (resp. local) modules is strongly lifting.
TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS
Shahabaddin Ebrahimi Atani,Saboura Dolati Pish Hesari,Mehdi Khoramdel 대한수학회 2014 대한수학회지 Vol.51 No.3
Let R be an I-semiring and S(R) be the set of all identity- summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(Γ(R)), and investigate basic properties of S(R) which help us to gain interesting results about T(Γ(R)) and its subgraphs.
TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL
Atani, Shahabaddin Ebrahimi,Hesari, Saboura Dolati Pish,Khoramdel, Mehdi Korean Mathematical Society 2015 대한수학회지 Vol.52 No.1
Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x, $y{\in}R$, the vertices x and y are adjacent if and only if $xy{\in}S(I)$.
THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS
Shahabaddin Ebrahimi Atani,Saboura Dolati Pish Hesari,Mehdi Khoramdel 대한수학회 2014 대한수학회지 Vol.51 No.1
An element r of a commutative semiring R with identity is said to be identity-summand if there exists 1 ≠ α ∈ R such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by Γ(R). It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph Γ(R) are studied.
THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS
Atani, Shahabaddin Ebrahimi,Hesari, Saboura Dolati Pish,Khoramdel, Mehdi Korean Mathematical Society 2014 대한수학회지 Vol.51 No.1
An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.
TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS
Atani, Shahabaddin Ebrahimi,Hesari, Saboura Dolati Pish,Khoramdel, Mehdi Korean Mathematical Society 2014 대한수학회지 Vol.51 No.3
Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(${\Gamma}(R)$), and investigate basic properties of S(R) which help us to gain interesting results about T(${\Gamma}(R)$) and its subgraphs.
TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL
Shahabaddin Ebrahimi Atani,Saboura Dolati Pish Hesari,Mehdi Khoramdel 대한수학회 2015 대한수학회지 Vol.52 No.1
Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if xy ∈ S(I).