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Let R be a finite ring with identity, T its set of idempotents. We study the subsets of T that can be closed under multiplication and the implications that fact has to the structure of R. We consider the subset M of all minimal idempotents and zero and prove that M is closed under multiplication if and only if R is a direct sum of local rings. We achieve this by studying the properties of units that are preserved by idempotents.

In this paper we study zero-divisor graphs of rings and semirings. We show that all zerodivisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle. In this paper we investigate the interplay between the algebraic properties of a (semi)ring and the graph theoretic properties of its zero-divisor graph. In the next section, we give all necessary definitions. In Sec. 3, we survey some of the known results of the theory of the zero-divisor graphs over semigroups, rings and semirings, and extend these results to a more general setting of a noncommutative semiring and we characterize all acyclic zero-divisor graphs of semirings (Theorem 10). Next, we study the cyclic zero-divisor graphs. First, we characterize the complete kpartite and regular zero-divisor graphs that can appear as the zero-divisor graphs of commutative semirings (Theorem 12 and Corollary 13). In the case the zero-divisor graph of a commutative semiring contains at most one triangle, we find all possible zero-divisor graphs (Theorems 20 and 26 and Proposition 22). If the zero-divisor graph of a commutative semiring is cyclic and contains no triangles, we describe the order of the nilpotent elements in the semiring (Proposition 21). In the case the zero-divisor graph of an additively cancellative semiring contains exactly one triangle, we prove that the semiring has to be a ring (Proposition 28) and we then proceed to characterize all rings and their zero-divisor graphs containing exactly one triangle (Theorem 32). DefinitionsA semiring is a set S equipped with binary operations + and · such that (S, +) is a commutative monoid with identity element 0, and (S, ·) is a monoid with identity element 1. In addition, operations + and · are connected by distributivity and 0 annihilates S. A semiring is commutative if ab = ba for all a, b ∈ S. A semiring is entire (or zero-divisor-free) if ab = 0 implies that a = 0 or b = 0. The semiring S is additively cancellative if a + c = b + c implies that a = b for all a, b, c ∈ S.The simplest example of a commutative semiring is the binary Boolean semiring, the set {0, 1} in which 1 + 1 = 1 · 1 = 1. We denote the binary Boolean semiring by B. Moreover, the set of nonnegative integers (or reals) with the usual operations of addition and multiplication, is a commutative semiring. Other examples of commutative semirings are distributive lattices, tropical semirings etc.

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In this paper we find all finite rings with a nilpotent group of units. It was thought that the answer to this was already given by McDonald in 1974, but as was shown by Groza in 1989, the conclusions that had been reached there do not hold. Here, we improve some results of Groza and describe the structure of an arbitrary finite ring with a nilpotent group of units, thus solving McDonald's problem.2000 Mathematics subject classification: primary 16P10; secondary 16U60.