Rings whose maximal one-sided ideals are two-sided

Abstract

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let $R$ be a ring and $R[x]$ be the polynomial ring over $R$ with $x$ the indeterminate. We show that $eRe$ is right quasi-duo for $0\neq e^2=e\in R$ if $R$ is right quasi-duo; $R/J(R)$ is commutative with $J(R)$ the Jacobson radical of $R$ if $R[x]$ is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if $R[x]$ is right quasi-duo then the Jacobson radical of $R[x]$ is $N(R)[x]$ and so the K\"othe's conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where $N(R)$ is the set of all nilpotent elements in $R$. Next we prove that if the polynomial ring $R[X]$, over a reduced ring $R$ with $|X|\geq 2$, is right quasi-duo, then $R$ is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.