NILPOTENT ELEMENTS AND NIL-ARMENDARIZ PROPERTY OF MONOID RINGS

作者：Habibi M; Moussavi A^*

来源：Journal of Algebra and Its Applications, 2012, 11(4): 1250080.

DOI：10.1142/S0219498812500806

• M-Armendariz rings
• Nil M-Armendariz rings
• 2-primal rings
• NI rings

Antoine [Nilpotent elements and Armendariz rings, J. Algebra 319(8) (2008) 3128-3140] studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. For a monoid M, we introduce nil-Armendariz rings relative to M, which is a generalization of nil-Armendariz rings and we investigate their properties. This condition is strongly connected to the question of whether or not a monoid ring R[M] over a nil ring R is nil, which is related to a question of Amitsur [Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956) 35-48]. This is true for any 2-primal ring R and u.p.-monoid M. If the set of nilpotent elements of a ring R forms an ideal, then R is nil-Armendariz relative to any u.p.-monoid M. Also, for any monoid M with an element of infinite order, M-Armendariz rings are nil M-Armendariz. When R is a 2-primal ring, then R[x] and R[x, x(-1)] are nil-Armendariz relative to any u.p.-monoid M, and we have nil(R[M]) = nil(R)[M].

• 出版日期2012-8