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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.
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참고문헌 (31)
참고문헌 신청
S. A. Amitsur / 1954 / A general theory of radicals III / Amer. J. Math 76:126~136
S. A. Amitsur / 1956 / Radicals of polynomial rings / Canad. J. Math 8:355~361
D. D. Anderson / 1998 / Armendariz rings and Gaussian rings / Comm. Algebra 26(7):2265~2272
R. Antoine / 2008 / Nilpotent elements and Armendariz rings / J. Algebra 319(8):3128~3140
R. Antoine / 2010 / Examples of Armendariz rings / Comm. Algebra 38(11):4130~4143
S. A. Amitsur / 1956 / Radicals of polynomial rings / Canad. J. Math 8:355~361
D. D. Anderson / 1998 / Armendariz rings and Gaussian rings / Comm. Algebra 26(7):2265~2272
R. Antoine / 2008 / Nilpotent elements and Armendariz rings / J. Algebra 319(8):3128~3140
R. Antoine / 2010 / Examples of Armendariz rings / Comm. Algebra 38(11):4130~4143
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