Abstract

An element $a$ in a non-commutative ring $R$ is called a strongly zero-divisor in $R$ if, either $\langle a \rangle\langle b \rangle=0$ or $\langle b \rangle\langle a \rangle=0$, for some $0\neq b\in R$ ($\langle x \rangle$ is the ideal generated by $x\in R$). This notion is introduced and extensively studied. Let $S(R)$ denotes the set of all strongly zero-divisors of $R$. For every ring $R$, we associate an undirected graph $\widetilde{\Gamma}(R)$ with vertices $S(R)^*:=S(R)\setminus \{0\}$, where distinct vertices $a$ and $b$ are adjacent if and only if either $\langle a \rangle\langle b \rangle=0$ or $\langle b \rangle\langle a \rangle=0$. We investigate the interplay of between the ring-theoretic properties of $R$ and the graph-theoretic properties of $\widetilde{\Gamma}(R)$. The graphs on $n=1, 2, 3$, or $4$ vertices which can be realized as the zero-divisor graphs of a commutative rings with 1, and a complete list of rings (up to isomorphism) producing these graphs have appeared in [Lecture Notes in Pure and Applied Mathematics, vol. 202, Marcel Dekker, NewYork, 2001, pp. 61-72]. This list was extended to $n=5$ vertices in [Comm. Algebra 31 (9) (2003) 4425-4443], and to $n=6, 7,...., 14$ vertices (for any commutative ring with 1) in [Discrete Mathematics 307 (2007) 1155-1166]. In this paper, the theory of (strong) zero-divisor graphs is applied to classification up to isomorphism of finite rings, and the structure and classification up to isomorphism of all finite rings (not necessary commutative or with identity) with at most fifteen zero-divisors are determined..



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