The Laskerian property in commutative rings

Abstract

1. INTRODUCTION Primary decomposition is a venerable tool in commutative algebra; indeed, Emmy Noether studied rings with the ascending chain condition on ideals because primary decomposition was available there [9 J. Though many results for which it was once used are now proved by other means, primary decomposition itself is still finding new applications [ 15, 161, and provides an often informative representation of ideals [2]. In this paper we study the class of rings (always commutative with unity) in which primary decom- position holds, and related classes. Recall: DEFINITION. Let M be a finitely generated module over ring R. (1) A submodule N is primary if, for any r in R and m in M whose product rm is in N, either m E N or some power rk of r satisfies rkM G N. It is strongly primary if, in addition, the radical P = fl= {r E R : rkM L N for some k} has a power Pk which satisfies PkM E N. (2) M is a (strongly) Laskerian module if every submodule of M is an intersection of a finite number of (strongly) primary submodules. (3) M is a ZD module if, for every submodule N of M, the set Z,(M/N) = {r E R: rm E N for some m E M\N} of zero divisors on M/N in R is the union of a finite number prime ideals in R. Of course, a ring is Laskerian, or strongly Laskerian, or ZD, if it has the property as a module over itself. In Section 2 we prove the ascent of these properties in certain ring extensions; in particular, finite integral extensions.