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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 I be a two-dimensional squarefree monomial ideal of a polynomial ring S. We evaluate the geometric regularity, a i -invariants of S / I n for i ≥ 2 . It turns out that they are all linear functions in n from n = 2 . Also, it is shown that g-reg (S / I n) = reg (S / I (n)) for all n ≥ 1 . [ABSTRACT FROM AUTHOR]

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]

Let R be a commutative Noetherian ring, a an ideal of R, M a finitely generated R-module, and S a Serre subcategory of the category of R-modules. We introduce the concept of S-minimax R-modules and the notion of the S-finiteness dimension faS(M):=inf{faRp(Mp)|p∈SuppR(M/aM)andR/p∉S}and we will prove that: (i) If Ha0(M),⋯,Han-1(M) are S-minimax, then the set {p∈AssR(Han(M))|R/p∉S} is finite. This generalizes the main results of Brodmann-Lashgari (Proc Am Math Soc 128(10):2851-2853, 2000), Quy (Proc Am Math Soc 138:1965-1968, 2010), Bahmanpour-Naghipour (Proc Math Soc 136:2359-2363, 2008), Asadollahi-Naghipour (Commun Algebra 43:953-958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860-4872, 2015). (ii) If S satisfies the condition Ca, then faS(M)=inf{i∈N0|Hai(M)is notS-minimax}.This is a formulation of Faltings’ Local-global principle for the S-minimax local cohomology modules. (iii) sup{i∈N0|Hai(M)is notS-minimax}=sup{i∈N0|Hai(M)is not inS}. [ABSTRACT FROM AUTHOR]

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and only if every R-module is serial. This motivated us to study commutative rings for which every proper ideal is serial. In this paper, we determine completely the structure of commutative rings R of which every proper ideal is serial. It is shown that every proper ideal of R is serial, if and only if, either R is a serial ring, or R is a local ring with maximal ideal ${\mathcal {M}}$ such that there exist a uniserial module U and a semisimple module T with ${\mathcal {M}}=U\oplus T$ . Moreover, in the latter case, every proper ideal of R is isomorphic to $U^{\prime }\oplus T^{\prime }$ , for some $U^{\prime }\leq U$ and $T^{\prime }\leq T$ . Furthermore, it is shown that every proper ideal of a commutative Noetherian ring R is serial, if and only if, either R is a finite direct product of discrete valuation domains and local Artinian principal ideal rings, or R is a local ring with maximal ideal ${\mathcal {M}}$ containing a set of elements { w ,..., w } such that ${\mathcal {M}}=\bigoplus _{i=1}^{n} Rw_{i}$ with at most one non-simple summand. Moreover, another equivalent condition states that: there exists an integer n ≥ 1 such that every proper ideal of R is a direct sum of at most n uniserial R-modules. Finally, we discuss some examples to illustrate our results. [ABSTRACT FROM AUTHOR]