Integral primary ideals of commutative rings

Abstract

Let $R$ be a commutative ring with identity. We call a proper ideal $I$ of $R$ integral primary if whenever $a,b\in R$ such that $ab\in I$, then either $a\in I$ or $b\in\overline{I}$ where $\overline{I}$ denotes the integral closure of $I$. The main purpose of this study is to give new characterizations for some special rings such as pr\'esimplifiable rings, $UN$-rings, von Neumann regular rings and fields by using the concept of integral primary ideals. Furthermore, we apply some properties of integral primary ideals to conclude the integral primary avoidance theorem.