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 $R$ of Goldie dimension 1 has Artinian classical quotient ring $Q$, hence is a Kerr ring in the sense that the polynomial ring $R[X]$ satisfies the $acc$ on annihilators $(=acc \bot )$. More generally, we show that a Goldie ring $R$ has Artinian $Q$ when every zero divisor of $R$ has essential annihilator (in this case $Q$ is a local ring; see Theorem $1^\prime$). A corollary to the proof is Theorem 2: A commutative ring $R$ has Artinian $Q$ iff $R$ is a Goldie ring in which each element of the Jacobson radical of $Q$ has essential annihilator. Applying a theorem of Beck we show that any $acc \bot$ ring $R$ that has Noetherian local ring $R_p$ for each associated prime $P$ is a Kerr ring and has Kerr polynomial ring $R[X]$ (Theorem 5).