Generalizations of $n$-Absorbing Ideals of Commutative Semirings

Pattarawan Petchkaew, Amorn Wasanawichit, Sajee Pianskool

Abstract

Let $R$ be a commutative semiring with nonzero identity and $\phi$  a function from $\mathscr{I}(R)$ into $\mathscr{I}(R) \cup \{\emptyset\}$ where $\mathscr{I} (R)$ is the set of ideals of $R$. Let $n$ be a positive integer. In this paper, we introduce the concept of $\phi$-$n$-absorbing ideals which are a generalization of $n$-absorbing ideals. A proper ideal $I$ of $R$ is called a $\phi$-$n$-absorbing ideal if whenever  $x_{1}x_{2}\cdots x_{n+1} \in I-\phi(I)$ for $x_{1},x_{2},\ldots,x_{n+1} \in R$, then  $x_{1}x_{2}\cdots x_{i-1}x_{i+1}\cdots x_{n+1}\in I$ for some $i \in \{1,2,\ldots,n+1\}$. A number of results concerning relationships between $\phi$-$n$-absorbing ideals and $n$-absorbing ideals as well as examples of $n$-absorbing ideals are given. Moreover, $\phi$-$n$-absorbing ideals of decomposable  semirings, of quotient semirings and of semirings of fractions are investigated.