A COUNTEREXAMPLE CONCERNING ACCP IN POWER SERIES RINGS

Abstract

ABSTRACT Several authors 1-4 studied unitary modules over (commutative) rings with unity that satisfy the ascending chain condition on n -generated submodules (“n-acc”) for a positive integer n . In the case of rings, the ascending chain condition on principal ideals (often called accp instead of 1-acc) is of special interest. W. Heinzer and D. Lantz in[2] described an example that accp does not rise to the polynomial ring in general, though in[1], they proved that the polynomial ring of a quasilocal accp-ring with only finitely many Bourbaki-weakly associated primes also satisfies accp. It is now a natural question if there is an analogous counterexample for power series rings. It is well-known and easy to see that for a quasilocal ring or an integral domain with accp, the power series ring in any number of indeterminates also satisfies accp (more generally, it suffices for the ring to have the property that every zero-divisor is in the Jacobson radical). We now give an example that accp does not rise to the power series ring in general. Further we show that for an accp-ring R every free R -module has 1-acc, a fact that was only known for integral domains untill now (Proposition 1.4 in[3]). This answers the case n=1 of the general question posed in[3]: Does every free module over a ring with n -acc also satisfy n -acc? In this paper, all rings are commutative with unity and all modules are unitary. We use the symbol for proper inclusion.