Bull. Korean Math. Soc. 2002; 39(3): 411-422
Printed September 1, 2002
Copyright © The Korean Mathematical Society.
Chan Huh, Sung Hee Jang, Chol On Kim, and Yang Lee
Pusan National University, Pusan National University, Pusan National University, Pusan National University
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.
Keywords: quasi-duo ring, polynomial ring, Jacobson radical, commutative ring
MSC numbers: Primary 16D60, 16S36; Secondary \linebreak 16D25, 16N20
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