On weak nil-armendariz rings
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2009
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Maltepe Üniversitesi
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Abstract
Rege and Chhawchharia introduce the notion of an Armendariz ring. A ring R is Armendariz if whenever f(x)g(x) = 0 where f(x) = a0 + a1x + · · · + anx n and g(x) = b0 + b1x + · · · + bmxm ? R[x], then aibj = 0 for each i and j. The name of the ring was given due to E. Armendariz who proved that reduced rings (i.e. rings without nonzero nilpotent elements) satisfied this condition. Armendariz rings are thus a generalization of reduced rings, and therefore, nilpotent elements play an important role in this class of rings. There are many examples of rings with nilpotent elements which are Armendariz. A ring is weak Armendariz if whenever the product of two polynomials is zero then the product of their coefficients is nilpotent. This further motivates the study of the nilpotent elements in this class of rings. We call a ring R weak nil-Armendariz if whenever f(x)g(x) ? nil(R)[x] where f(x) = a0 + a1x + · · · + anx n and g(x) = b0 + b1x + · · · + bmxm ? R[x], then a0bj ? nil(R) for each j. We prove that if R is a nil-Armendariz ring, then the set of nilpotent elements of R is a subring without unit of R. This allows us to study the conditions under which the polynomial ring over a nil-Armendariz ring is also nil-Armendariz. These conditions are strongly connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil.
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International Conference of Mathematical Sciences
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Hashemi, E. (2009). On weak nil-armendariz rings. Maltepe Üniversitesi. s. 146.
