Classical quotient rings of PWD’s

Abstract

Piecewise domains which are right orders in semiprimary rings are characterized. An example is given showing the result obtained is best possible. A further example is obtained of a prime right Goldie ring possessing a regular element which becomes a left zero divisor in some prime overring. This example leads to the construction of a PWD R not satisfying the regularity condition, but for which R/N(R) is right Goldie. Introduction. The principle result of this note is a characterization of PWD's (piecewise domains) which are right orders in semiprimary rings. The main device employed here is L. W. Small's concept of exhaustive set. Probably the most important consequence of the characterization is that a right noetherian PWD is a right order in a right artinian PWD. In the more general setting our result is less satisfactory because an assumption is made about each of the factor rings R/Ti(R) (see the theorem in ?1). However, an example given at the end of ?1 shows that this is unavoidable: There we exhibit a right Goldie PWD R having R/N(R) right Goldie which is not a right order in a semiprimary ring. In ?2, as a problem closely related to the existence of classical quotient rings, we study the regularity condition in PWD's. Necessary and sufficient conditions are obtained for a PWD R to satisfy the regularity condition in terms of the torsion-freeness of R as an R/N(R)-module. It is shown that any PWD R with R/N(R) right Goldie satisfies a certain one-sided regularity condition. That the regularity condition then need not be implied is demonstrated by an example. Since this example is surely of more general interest than its application, we single it out here: A right Ore domain D is constructed so that for some elements a, b, c, d E D, the equations ax+by=O, cx+dy=O have a nontrivial solution in a domain E extending D, but no nontrivial solution in D. (This means that the prime right Goldie ring D2 contains a regular element which becomes a left zero divisor in the prime overring E2.) It is amusing to note that there Received by the editors September 7, 1971. AMS 1970 subject classifications. Primary 16A08; Secondary 16A02, 15A06. Kev words and phrases. Piecewise domain, essentiality condition, exhaustive set, order, classical quotient ring, Goldie ring, principal right ideals projective, torsionfree, left regularity condition, Ore domain, linear equations. ( American Mathematical Society 1972