On Armendariz rings (Q839699)

Let \(R\) be a commutative ring with identity. For a polynomial \( f\in R[X]\) the \textit{content} of \(f\) is the ideal \(C(f)\) of \(R\) generated by the coefficients of \(f\). The ring \(R\) is said to be \textit{Armendariz} if whenever \(f,g\in R[X]\) with \(fg=0\), then \(C(f)C(g)=0\) [\textit{M. B.\ Rege} and \textit{S. Chhawachharia}, Proc. Japan Acad., Ser. A 73, No. 1, 14--17 (1997; Zbl 0960.16038)]. It is shown that the trivial extension or idealization \(R(+)M \) where \(R\) is quasi-local and \(M\) is a semisimple \(R\)-module is Armendariz if and only if \(R\) is Armendariz and that if \(A\subseteq B\)\ is an extension of integral domains, then \(A(+)B\) is Armendariz. The paper also shows that a direct product of rings is Armendariz if and only if each factor is that a ring \(R\) is Armendariz if and only if \(R_{M}\) is Armendariz for each maximal ideal \(M\) of \(R\). The second result on idealization in a more general form and the last two results are given in [\textit{D. D. Anderson} and \textit{V.\ Camillo}, Commun. Algebra 26, No. 7, 2265--2272 (1998; Zbl 0915.13001)].