ON JACOBSON AND NIL RADICALS RELATED TO POLYNOMIAL RINGS

• Published in: Journal of the Korean Mathematical Society Vol. 53; no. 2; pp. 415 - 431
• Main Authors: Kwak, Tai Keun, Lee, Yang, Ozcan, A. Cigdem
• Format: Journal Article
• Language: English
• Published: 대한수학회 01.03.2016
• Subjects: 수학
• ISSN: 0304-9914; 2234-3008
• DOI: 10.4134/JKMS.2016.53.2.415

This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that $J(R[x])=J(R)[x]$ if and only if $J(R)$ is nil when a given ring $R$ is Armendariz, where $J(A)$ means the Jacobson radical of a ring $A$. A ring will be called {\it feckly Armendariz} if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings. KCI Citation Count: 4