# RING WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED

• Huh, Chan (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
• Jang, Sung-Hee (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
• Kim, Chol-On (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
• Lee, Yang (DEPARTMENT OF MATHEMATICS EDUCATION, PUSAN NATIONAL UNIVERSITY)
• Published : 2002.08.01

#### 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[$\chi$] 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 $K\ddot{o}the's$ 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 rins R[x], over a reduced ring R with $\mid$X$\mid$ $\geq$ 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.

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#### Cited by

1. Quasi-duo skew polynomial rings vol.212, pp.8, 2008, https://doi.org/10.1016/j.jpaa.2008.01.002