Strongly regular elements of noetherian rings

Abstract

Any regular element of a commutative ring remains regular modulo every annihilator ideal of the ring. In general, this is not the case if the assumption of commutativity is dropped. Although certain elements, for example the units, stay regular modulo every right or left annihilator ideal, this may not be the case for other regular elements. Thus the multiplicatively closed set Y(O) of all those regular elements which are well behaved in this sense deserves some special attention, its elements are called strolzgly regular. This set is particularly interesting for questions concerning quotient rings, for it turns out that in a right noetherian ring any regular right Ore set is contained in Y(O) (Proposition 1.6). Thus the equality of -i”(O) and g(O), the set of regular elements, becomes a necessary condition for the existence of a full right quotient ring. For a (left and right) noetherian ring R it turns out that Y(O) = n g(P), where P runs through the set of all prime middle annihilator ideals (Theorem 2.1). A similar description of G?(O) in terms of finitely many affiliated primes obtained by L. Small and J. T. Stafford [8] thus makes the comparison of the two sets relatively easy. Although the inequality of Y(O) and @Y(O) provides a criterion for ruling out a full right quotient ring, equality of the two sets by no means guarantees its existence (Example 2.6).