A note on nil power serieswise Armendariz rings.
Let \(R\) be a ring with 1, \(R[x]\) a polynomial ring, \(R[\![x]\!]\) the formal power series ring with an indeterminate \(x\), \(\text{Nil}(R)\) the nil radical of \(R\). A ring is called weak (resp. nil) Armendariz if for \(f=\sum_{i=0}^na_ix^i\) and \(g=\sum_{i=0}^mb_ix^i\in R[x]\) such that \(fg=0\) (resp. \(fg\in\text{Nil}(R)[x]\)), then \(a_ib_j\in\text{Nil}(R)\) for all \(i\) and \(j\). A ring is called weak (resp. nil) power serieswise Armendariz if for \(f=\sum a_ix^i\) and \(g=\sum b_ix^i\in R[\![x]\!]\) such that \(fg=0\) (resp. \(fg\in\text{Nil}(R)[\![x]\!]\)), then \(a_ib_j\in\text{Nil}(R)\) for all \(i\) and \(j\). Then the author gives some equivalent conditions for a nil power serieswise Armendariz ring. Theorem 1. The following statements are equivalent: (1) \(R\) is a nil power serieswise Armendariz ring; (2) \(\text{Nil}(R)\) is an ideal of \(R\); (3) For an ideal \(I\subset\text{Nil}(R)\), \(R/I\) is nil power serieswise Armendariz; (4) For any positive integer \(n\), \(R[x]/\langle x^n\rangle\) is nil power serieswise Armendariz. Theorem 2. Let \(R\) be nil power serieswise Armendariz. Then the following statements are equivalent: (1) \(R[x]\) is nil power serieswise Armendariz; (2) \(\text{Nil}(R[x])=\text{Nil}(R)[x]\); (3) \(\text{Nil}(R)[x]\subset\text{Nil}(R[x])\); (4) \(R[x]\) is nil Armendariz; (5) \(R[x,x^{-1}]\) is nil Armendariz; (6) \(R[x,x^{-1}]\) is nil power serieswise Armendariz. -- Moreover, if \(R\) is nil power serieswise Armendariz of bounded index of nilpotency, then \(R[\![x]\!]\) is nil power serieswise Armendariz.
scientific article; zbMATH DE number 5723774