In this paper, we introduce a new graph whose vertices are the non-zero zero-divisors of a commutative ring R, and for distincts elements x and y in the set Z(R)* of the non-zero zero-divisors of R, x and y are adjacent if and only if xy = 0 or x + y ∈ Z(R). We present some properties and examples of this graph, and we study its relationship with the zero-divisor graph and with a subgraph of the total graph of a commutative ring.

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It is known that the socle of a semiprime Goldie ring is generated by a central idempotent and that a prime Goldie ring with a nonzero socle is a simple artinian ring. We prove the extension of these results to alternative rings. We also give an analogue of Goldie's theorem for alternative rings. A Goldielike theorem was obtained earlier by the authors for noetherian alternative rings by a quite different method.

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