Abstract
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 of a commutative semiring with respect to the singular ideal. We define this graph as the undirected graph T(Γ(S)) with all elements of S as vertices and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z(S). We discuss various characteristics of this total graph and also characterize some important properties of certain induced subgraphs of this total graph.
Highlights
In the last two decades, the study of algebraic structures using the properties of graphs has emerged as an exciting area of research
This study began way back in 1988 with Istvan Beck establishing a correspondence between the theory of graph and ring theory
In his paper [18], Beck introduced the notion of coloring of a commutative ring by defining the zero-divisor graph of a commutative ring
Summary
In the last two decades, the study of algebraic structures using the properties of graphs has emerged as an exciting area of research. For a commutative ring R, they defined this graph as the undirected graph with R as the vertex set and if x and y are any two vertices they are adjacent if and only if x + y ∈ Z(R), where Z(R) denotes the set of zero divisors of R.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
