Polynomial rings over Goldie-Kerr commutative rings II

Abstract

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989–993) is that any Goldie ring R R of Goldie dimension 1 has Artinian classical quotient ring Q Q , hence is a Kerr ring in the sense that the polynomial ring R [ X ] R[X] satisfies the a c c acc on annihilators ( = a c c ⊥ ) (=acc \bot ) . More generally, we show that a Goldie ring R R has Artinian Q Q when every zero divisor of R R has essential annihilator (in this case Q Q is a local ring; see Theorem 1 ′ 1^\prime ). A corollary to the proof is Theorem 2: A commutative ring R R has Artinian Q Q iff R R is a Goldie ring in which each element of the Jacobson radical of Q Q has essential annihilator. Applying a theorem of Beck we show that any a c c ⊥ acc \bot ring R R that has Noetherian local ring R p R_p for each associated prime P P is a Kerr ring and has Kerr polynomial ring R [ X ] R[X] (Theorem 5).