ASSOCIATED PRIMES OVER SKEW POLYNOMIAL RINGS

Abstract

In this paper, we extend the recent work of C. FaithCitation[1] in which it is proved, using results of R.C. Shock,Citation[2] that over a commutative ring , the associated primes of the polynomial ring (viewed as a module over itself) are all extended; that is, every may be expressed as , where . This result was originally proved in 1974 by Brewer and Heinzer using localization theory (seeCitation[3]), but our work in this paper more closely parallels that of Faith. We will show that the result still holds in the more general setting of a polynomial module over a skew polynomial ring , with possibly noncommutative base ring , provided a natural -compatibility assumption on (as defined in Sec. 2 below) is fulfilled. Moreover, if we assume that , we can make the set of Laurent polynomials into a module over and , and the main result still holds in these cases as well (for -compatible ). Finally, we also show that the result does not extend to the case of skew formal power series rings and skew Laurent series rings in general, although it does if is a left perfect ring.