RING WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED

Title & Authors
RING WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED
Huh, Chan; Jang, Sung-Hee; Kim, Chol-On; Lee, Yang;

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 $\small{0{\neq}e^2=e{\in}R}$ if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[$\small{\chi}$] 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 $\small{K\ddot{o}the$ 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 $\small{\mid}$X$\small{\mid}$ $\small{\geq}$ 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
Language
English
Cited by
1.
ON FULLY IDEMPOTENT RINGS,;;;

대한수학회보, 2010. vol.47. 4, pp.715-726
1.
Quasi-duo skew polynomial rings, Journal of Pure and Applied Algebra, 2008, 212, 8, 1951
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