Abstract
The paper studies the following question: Given a ring R , when does the zero-divisor graph Γ ( R ) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then Γ ( R ) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z 2 × Z 2 × Z 2 ; Z 2 × Z 4 ; Z 2 × ( Z 2 [ x ] / ( x 2 ) ) ; F 1 × F 2 , where F 1 , F 2 are fields. In addition, we determine all positive integers n for which Γ ( Z n ) has the property.
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