On rings of fractions

Let R be a commutative Noetherian ring with identity, and let M be a fixed ideal of R . Then, trivially, ring multiplication is continuous in the ilf-adic topology. Let S be a multiplicative system in R , and let j = j s : R → S -1 R, be the natural map. One can then ask whether (cf. Warner [3, p. 165]) S -1 R is a topological ring in ihe j ( M )-adic topology. In Proposition 1, I prove this is the case if and only if M ⊂ p ( S ), where Hence S -1 R is a topological ring for all S if and only if M ⊂ p*(R), where