Polynomial Rings Over Goldie-Kerr Commutative Rings

Abstract

All rings in this paper are commutative, and $\operatorname {acc} \bot$ (resp., $\operatorname {acc} \oplus$) denotes the acc on annihilators (resp., on direct sums of ideals). Any subring of an $\operatorname {acc} \bot$ ring, e.g., of a Noetherian ring, is an $\operatorname {acc} \bot$ ring. Together, $\operatorname {acc} \bot$ and $\operatorname {acc} \oplus$ constitute the requirement for a ring to be a Goldie ring. Moreover, a ring $R$ is Goldie iff its classical quotient ring $Q$ is Goldie. A ring $R$ is a Kerr ring (the appellation is for J. Kerr, who in 1990 constructed the first Goldie rings not Kerr) iff the polynomial ring $R[x]$ has $\operatorname {acc} \bot$ (in which case $R$ must have $\operatorname {acc} \bot$). By the Hilbert Basis theorem, if $S$ is a Noetherian ring, then so is $S[x]$; hence, any subring $R$ of a Noetherian ring is Kerr. In this note, using results of Levitzki, Herstein, Small, and the author, we show that any Goldie ring $R$ such that $Q = {Q_c}(R)$ has nil Jacobson radical (equivalently, the nil radical of $R$ is an intersection of associated prime ideals) is Kerr in a very strong sense: $Q$ is Artinian and, hence, Noetherian (Theorems 1.1 and 2.2). As a corollary we prove that any Goldie ring $A$ that is algebraic over a field $k$ is Artinian, and, hence, any order $R$ in $A$ is a Kerr ring (Theorem 2.5 and Corollary 2.6). The same is true of any algebra $A$ over a field $k$ of cardinality exceeding the dimension of $A$ (Corollary 2.7). Other Kerr rings are: reduced $\operatorname {acc} \bot$ rings and valuation rings with $\operatorname {acc} \bot$ (see 3.3 and 3.4).