Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph
By David F. Anderson, John D. LaGrange. 2012
65 citations recorded by the source corpus.
Let R be a commutative ring with 1≠0. The zero-divisor graph Γ(R) of R is the (undirected) graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. The relation on R given by r∼s if and only if annR(r)=annR(s) is an equivalence relation. The compressed zero-divisor graph ΓE(R) is the (undirected) graph whose vertices are the equivalence classes induced by ∼ other than [0] and [1], such that distinct vertices [r] and [s] are adjacent in ΓE(R) if and only if rs=0. We investigate ΓE(R) when R is reduced and are interested in when ΓE(R)≅Γ(S) for a reduced ring S. Among other results, it is shown that ΓE(R)≅Γ(B) for some Boolean ring B if and only if Γ(R) (and hence ΓE(R)) is a complemented graph, and this is equivalent to the total quotient ring of R being a von Neumann regular ring.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/2a8f8ecb4b6ba384c0f495fbb682789dc08c1421
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In Algebra and Number Theory
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Citations in the corpus
- On the Dot Product Graph of a Commutative Ring (2015, unknown): “Recently, there has been considerable attention in the literature to associating graphs with algebraic structures (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [23], [24], [25], and [26]).”
- On Zero Divisor Graphs (2012, methodology): “(In [8], this is called the “compressed” zero divisor graph.”