ANNIHILATOR IDEALS, ASSOCIATED PRIMES AND KASCH-MCCOY COMMUTATIVE RINGS

If R is a ring, and A right annulet ( = annihilator right ideal) then A[X] is a right annulet of the polynomial ring R[X]. (In fact, X can be any set of variables). An annulet I of R[X] of this form is said to be extended. Not all annulets of R[X] are extended, since, e.g., the ascending chain on right annulets (= acc perpendicular-to) is not inherited by R[X], as Kerr [Ke] observed. Nevertheless, maximal (minimal) annulets of a polynomial ring R[X] are extended, as a theorem of McCoy on annihilators in R[X] readily shows (see Introduction). Recall that a maximal proper annulet (= maxulet in the text) of a commutative ring R is a prime ideal, called an associated prime of R. The result just stated shows that the contraction map Ass R[X] --> c Ass R is an injection. We consider conditions which imply this is a bijection, i.e., that every maxulet of R extends to a maxulet of R[X]. A ring R has right E max if every right annulet ] R is contained in a right maxulet. Clearly, right E max is equivalent to the dual condition left E min, and for commutative R one has E max iff E min. One easily sees that a reduced (= semiprime) ring R has E min iff its maximal quotient ring is a product of fields. (Theorem 7.6) Our main results state for commutative R that Ass R[X] --> c Ass R is a bijection under any of the following assumptions: (1) R[X] has E max (Theorem 7.1). (2) R/P is Noetherian FOR-ALL-P is-an-element-of Ass R (Theorem 7.2). (3) R is a zip ring, i.e. has the finite intersection property for zero intersections of annihilators. (See text.) (Theorem 3.1) (4) A reduced ring R with E max (Corollary 7.7). Actually, by Theorem 3.11, a right zip ring has left E max. A theorem of Beachy and Blair implies that R[X] is zip iff R is, when R is commutative. We use these theorems to prove that R[X] has (semilocal) Kasch classical quotient ring Q(c) (R[X]) iff R does. We study subrings of rings that are completely zip in the sense that all factor rings are zip, and show that they must all be zip rings. A similar theorem holds if all factor rings have semilocal Kasch quotient rings. In the last section, we show that the finite Abelian group ring and full matrix ring over a commutative zip ring is zip.