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For a commutative ring R with zero-divisors Z (R), the zero-divisor graph of R is Γ (R) = Z (R) − {0}, with distinct vertices x and y adjacent if and only if x y = 0. In this paper, we characterize when either diam(Γ (R)) ≤ 2 or gr(Γ (R)) ≥ 4. We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.

ABSTRACT. Let Tbe a domain of the form K + M, where K is a field and M is a maximal ideal of T. Let D be a subring of K and let R = D + M. It is proved that if K is algebraic over D and both D and T are universally catenarian, then R is universally catenarian.The converse holds if K is the quotient field of D. As a consequence, we construct for each n > 2, an n-dimensional universally catenarian domain which does not belong to any previously known class of universally catenarian domains.

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