Abstract. We will state some conditions under which if a quotient of a module M satisfies the radical formula of degree k (s.t.r.f of degree k), so does M. Especially, we will introduce some particular modules M such that M ⊕ M s. Introduction.In this paper all rings are commutative and with identity, all modules are unitary, R denotes a ring and M denotes an R-module. Also, by ‫ގ‬ we mean the set of positive integers and ‫ގ‬ * = ‫ގ‬ ∪ {0, ∞}. A proper submodule P of M is called prime when from rm ∈ P for some r ∈ R and m ∈ M, we can conclude either m ∈ P or rM ⊆ P (see for example [1, 3, 4, 6, 9, 10, 11, 13, 15, 16]). Let (P : M) be the set of all r ∈ R such that rM ⊆ P. If P is a prime submodule, then P = (P : M) is a prime ideal of R and we say that P is P-prime. Recall that if R is an integral domain, then M is called torsion-free, if for every 0 = r ∈ R and 0 = m ∈ M, we have rm = 0. One can easily verify that P is a P-prime submodule of M if and only if

In this paper, we study the notion of chordality and cycles in hypergraphs from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. However, there is no unified definition for cycle or chordality in hypergraphs in the literature, so we consider several generalizations of these notions and study their algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of hypergraphs. Also we show that if C is a hypergraph such that C is a vertex decomposable simplicial complex or I(C) is squarefree stable, then C is chordal according to one of the most promising definitions.

In this paper, we consider factorizations of elements of a divisor-closed multiplicative submonoid of a ring and also factorizations of elements of a module as a product of elements coming from a divisor-closed multiplicative submonoid of the ring and another element of the module. In particular, we study uniqueness and some other properties of such factorizations and investigate the behavior of these factorizations under direct sum and product of rings and modules.

In this paper, we consider the generalization of chordal graphs to clutters proposed by Bigdeli, et al in J. Combin. Theory, Series A (2017). Assume that C is a d-dimensional uniform clutter. It is known that if C is chordal, then I(C) has a linear resolution over all fields. The converse has recently been rejected, but the following question which poses a weaker version of the converse is still open: "if I(C) has linear quotients, is C necessarily chordal?". Here, by introducing the concept of the ascent of a clutter, we split this question into two simpler questions and present some clues in support of an affirmative answer. In particular, we show that if I(C) is the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, then C is chordal.