On zero divisor graph of unique product monoid rings over Noetherian reversible ring

 Let R be an associative ring with identity and Z^*(R) be its set of non-zero zero divisors.  The zero-divisor graph of R, denoted by \Gamma(R), is the graph whose vertices are the non-zero  zero-divisors of  R, and two distinct vertices r and s are adjacent if and only if rs=۰ or sr=۰.  In this paper, we bring some results about undirected zero-divisor graph of a monoid ring over reversible right (or left) Noetherian ring R. We essentially classify the diameter-structure of this graph and show that ۰\leq \mbox{diam}(\Gamma(R))\leq \mbox{diam}(\Gamma(R[M]))\leq ۳. Moreover, we give a characterization for the possible diam(\Gamma(R)) and diam(\Gamma(R[M])), when R is a reversible Noetherian ring and M is a u.p.-monoid. Also, we study relations between the girth of \Gamma(R) and that of \Gamma(R[M]).

کلیدواژه ها:

Zero-divisor graphs ، diameter ، Girth ، Reversible rings ، Polynomial rings ، Unique product monoids ، Monoid rings