# TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS

@article{Atani2014TOTALGO, title={TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS}, author={Shahabaddin Ebrahimi Atani and Saboura Dolati Pish Hesari and M. Khoramdel}, journal={Journal of Korean Medical Science}, year={2014}, volume={51}, pages={593-607} }

Abstract. 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 graphof R with respect to identity-summand elements, denoted by T(Γ(R)),and investigate basic properties of S(R) which help us to gain interestingresults about T(Γ(R)) and its subgraphs. 1. IntroductionAssociating a graph to an algebraic structure is a research subject and hasattracted considerable attention. In fact, the research in this subject aims atexposing the…

## 9 Citations

### TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL

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In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and…

### Total graph of a commutative semiring with respect to singular ideal

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Let S be a commutative semiring with unity. The singular ideal Z(S) of S is defined as Z(S) = {s ? S | sK = 0 for some essential ideal K of S}. In this paper, we introduce the notion of total graph…

### A graph associated to a commutative semiring

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Let R be a commutative semiring with nonzero identity and H be an arbitrary multiplicatively closed subset R. The generalized identitysummand graph of R is the (simple) graph GH(R) with all elements…

### Total graph of a $0$-distributive lattice

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Let £ be a $0$-distributive lattice with the least element $0$, the greatest element $1$, and ${rm Z}(£)$ its set of zero-divisors. In this paper, we introduce the total graph of £, denoted by ${rm…

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Let R and S be two commutative rings with unity, let I be an ideal of S and $$\varphi : R \longrightarrow S$$ φ : R ⟶ S be a ring homomorphism. In this paper, we give a characterization for the the…

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The regular graph of ideals of the commutative ring $R$, denoted by $\Gamma_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$…

### More on the annihilator-ideal graph of a commutative ring

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Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see…

### Intersection graphs of co-ideals of semirings

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Let R be a semiring with identity. In this paper, we introduce the intersection graph of co-ideals, denoted by G(R). The vertices of G(R) are non-trivial co-ideals of R, and two distinct vertices I…

## References

SHOWING 1-10 OF 26 REFERENCES

### THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS

- Mathematics
- 2014

Abstract. An element r of a commutative semiring R with identity issaid to be identity-summand if there exists 1 6= a∈Rsuch that r+a= 1.In this paper, we introduce and investigate the…

### THE TOTAL GRAPH OF A COMMUTATIVE RING WITH RESPECT TO PROPER IDEALS

- Mathematics
- 2012

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect…

### THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-I

- Mathematics
- 2013

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R,…

### THE TOTAL TORSION ELEMENT GRAPH OF A MODULE OVER A COMMUTATIVE RING

- Mathematics
- 2011

The total graph of a commutative ring have been introduced and studied by D. F. Anderson and A. Badawi in [1]. In a manner analogous to a commutative ring, the total torsion element graph of a module…

### ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT

- Mathematics
- 2012

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct…

### ZERO-DIVISOR GRAPHS WITH RESPECT TO PRIMAL AND WEAKLY PRIMAL IDEALS

- Mathematics
- 2009

Abstract. We consider zero-divisor graphs with respect to primal, non-primal, weakly prime and weakly primal ideals of a commutative ring Rwith non-zero identity. We investigate the interplay between…

### ON THE ASSOCIATED GRAPHS TO A COMMUTATIVE RING

- Mathematics
- 2012

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by ΓS(R) with all elements of R as vertices, and two…