Nilpotent elements and Armendariz rings. (Q932855)

Let \(R\) denote an associative ring with \(1\), and let \(\text{nil}(R)\) denote the set of nilpotent elements. Further, let \(f(x)=\sum_{i=0}^ma_ix^i,g(x)=\sum_{j=0}^nb_jx^j\in R[x]\) denote two arbitrary polynomials. One says that \(R\) is an Armendariz ring if \(f(x)g(x)=0\) implies that \(a_ib_j=0\) for all \(i\) and \(j\). Similarly, one says that \(R\) is a weak Armendariz ring if \(f(x)g(x)=0\) implies \(a_ib_j\in\text{nil}(R)\) for all \(i\) and \(j\). The author makes the following definition, which is a mixture of these two: a ring \(R\) is nil-Armendariz if whenever \(f(x)g(x)\in\text{nil}(R)[x]\) then \(a_ib_j\in\text{nil}(R)\) for all \(i\) and \(j\). The author proves that Armendariz rings are nil-Armendariz, gives examples to show the converse implication does not hold, classifies when certain coproducts of algebras over fields are Armendariz, and proves that \(\text{nil}(R)\) is a subring when \(R\) is a (nil-)Armendariz ring. The author also proves that if \(R\) is nil-Armendariz then \(R/\text{Nil}^*(R)\) is Armendariz, but \(R/\text{Nil}_*(R)\) is not in general Armendariz. A number of interesting and clarifying examples are provided.