The Generalized Total Graph of a Commutative Ring

Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, Nil(R) be its ideal of nilpotent elements, and U(R) be its group of units. We define a nonempty proper subset H of R to be a multiplicative-prime subset of R if the following two conditions hold: (i) ab ∈ H for every a ∈ H and b ∈ R; (ii) if ab ∈ H for a, b ∈ R, then either a ∈ H or b ∈ H . For example, H is multiplicativeprime subset of R if H is a prime ideal of R, H is a union of prime ideals of R, H = Z(R), or H = R\U(R). In fact, it is easily seen that H is a multiplicativeprime subset of R if and only if R\H is a saturated multiplicatively closed subset of R. Thus H is a multiplicative-prime subset of R if and only if H is a union of