An Application of $SB$-Rings

Abstract

All rings are commutative rings with identity and $J(R)$ denotes the Jacobson radical of a ring $R$. A ring $R$ is called a $SB$-ring provided that for any sequence $a_1, a_2, \ldots , a_s, a_{s+1}$ of elements in $R$ with $s \ge 2$ and $(a_1, a_2, \ldots , a_{s-1} ) \not\subseteq J(R)$, there exists $b \in R$ such that $(a_1, a_2, \ldots a_s, a_{s+1}) = (a_1, a_2, \ldots , a_s + ba_{s+1})$. By applying some of the properties of $SB$-rings, it is shown that $R[X]$ is not a Prüfer domain for any Noetherian domain $R$ which is not a field.