Search

Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N :RM) or m ∈ M − rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

Let R be a standard graded algebra over an infinite field K and M a finitely generated Z -graded R -module. For any graded ideal I ⊆ R + of R , we show that the functions D ( I n M ) , r ( I n M ) and r ( M / I n M ) are all asymptotically linear. Here r ( • ) and D ( • ) stand for the reduction number and the maximal degree of minimal generators of a graded module • respectively. [ABSTRACT FROM AUTHOR]