Nil-Armendariz Condition on skew generalized power series rings

Document Type : Research Note

Authors

1 Payame Noor University

2 Tafresh University

Abstract

Let $R$ be a ring, $S$ a strictly ordered monoid, and $omega:Srightarrow End(R)$ a monoid homomorphism. The skew
generalized power series ring $R[[S,omega]]$ is a common generalization of (skew) polynomial rings, (skew)
power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent
series rings. In the current work, we study the
$(S,omega)$-nil Armendariz condition on $R$, a generalization of the standard nil-Armendariz
condition from polynomials and power series to skew generalized power series. We resolve the structure of $(S,omega)$-nil Armendariz rings
and obtain various necessary or sufficient conditions for a ring to be $(S,omega)$-nil Armendariz,
unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special
cases. For example, we show that left uniserial nilpotent semicomutative rings are nil Armendariz. Moreover, we
study on the relationship between the zip and weak zip properties of
a ring $R$ and these of the ring $R[[S,omega]]$.

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Main Subjects

Iranian Journal of Science

Articles in Press, Accepted Manuscript
Available Online from 04 October 2015

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