Let R be a commutative ring and M an R -module. In this article, we introduce the concept of S -2-absorbing submodule. Suppose that S ⊆ R is a multiplicatively closed subset of R. A submodule P of M with ( P : _R M ) ∩ S = ∅ is said to be an S -2-absorbing submodule if there exists an element s ∈ S and whenever abm ∈ P for some a, b ∈ R and m ∈ M , then sab ∈ ( P : _R M ) or sam ∈ P or sbm ∈ P. Many examples, characterizations and properties of S -2-absorbing submodules are given. Moreover, we use them to characterize von Neumann regular modules in the sense [9].

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. In summary, we give generalizations of $ (k,n) $-absorbing hyperideal and $ (k,n) $-absorbing primary hyperideal. Also, we examine the relationships 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.

Institute of Mathematics of the Czech Academy of Sciences