On strongly $1$-absorbing primary ideals of commutative rings

Abstract

Let $R$ be a commutative ring with $1\neq 0$. In this paper, we introduce a subclass of the class of $1$-absorbing primary ideals called the class of strongly $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly $1$-absorbing primary if whenever nonunit elements $a, b, c \in R$ and $abc \in I$, then $ab \in I$ or $c \in \sqrt{0}$. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the $UN$-rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.