Let $S$ be a commutative ring, and let $A$ be a faithful $S$-module. Through idealization, we can make the abelian group $R=S⊕A$ into a commutative ring using the multiplication $(s,a)(s',a')=(ss',sa'+s'a)$. By Problem 16, section D, Chapter 1 of the book Ring Theory. Nonsingular Rings and Modules by Kenneth Goodearl, the singular ideal $Z(R)=T⊕A$, where $T=\{s\in S\mid Bs=0$ for some essential submodule $B$ of $A$}. My question:
If $z\in Z(S)$, the singular ideal of $S$, and $a\in A$ what condition may be imposed on $z$ and/or $a$ to guarantee that $(z,a)\in Z(R)$?
From $z\in Z(S)$ we get that the annihilator ideal of $z$ in $S$ is essential in $S$ meaning that its intersection with any nonzero ideal of $S$ is nonzero; and I search for an essential submodule $B$ of $A$ which annihilates the element $z$. But I can't get any good idea for introducing $B$. Thanks for any cooperation!