In this paper, we introduce and study [Formula: see text]-1-absorbing prime ideals of commutative rings. Let [Formula: see text] be a ring and [Formula: see text] a positive integer. A proper ideal [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-1-absorbing prime ideal if whenever [Formula: see text] for some nonunits [Formula: see text] then either [Formula: see text] or [Formula: see text] It is obvious that 1-1-absorbing (2-1-absorbing) prime ideals are exactly prime (1-absorbing prime) ideals. Various examples and characterizations of [Formula: see text]-1-absorbing prime ideals are given.

For an ideal $I$ of a commutative ring $R$, we classify the set of all $3$-zero divisor elements of $R$ with respect to $I$. We prove that if $\bigcap_{i=1}^n P_i$ is a minimal prime decomposition of $\sqrt{I}$, then all elements of \begin{center} $\displaystyle{\bigcup_{i=1}^n P_i-\Big( \bigcup_{j=1}^n \Big\{x\in\bigcap_{i\in \{1,2,\ldots,n\}-\{j\}} P_i\backslash P_j: P_j\subseteq I:_Rx \Big\} \cup \sqrt{I}\Big)}$ \end{center} are $3$-zero divisor elements of $R$ with respect to $I$. Moreover, we determine $3$-zero divisors elements of $R$ with respect to $I$, when either $\sqrt{I}=I$ or $I$ is a primary ideal. Afterwards, we introduce quasi $3$-zero divisor element of $R$ with respect to $I$ and by using the minimal prime ideals of $R$ containing $I$ we characterize these elements. Moreover, we define quasi 3-zero divisor hypergraph of $R$ with respect to $I$ and we determine some properties of this hypergraph with the algebraic properties of $R$. 2000 Mathematics Subject Classification. 13A15, 13F30, 05C25.

In this paper, we introduce new expansion classes, namely weakly $ (k,n) $-absorbing hyperideals and weakly $ (k,n) $-absorbing primary hyperideals of a Krasner $ (m,n) $-hyperring, including $ (k,n) $-absorbing hyperideal and $ (k,n) $-absorbing primary hyperideal. Therefore, we give generalizations of $ (k,n) $-absorbing hyperideal and $ (k,n) $-absorbing primary hyperideal. Also, we examine the relations between classical hyperideals and the new hyperideals and explore some ways to connect them. Additionally, some main results and examples are given to explain the structures of these concepts. Finally, we study a version of Nakayama's lemma on a commutative Krasner $ (m,n) $-hyperring.