The Cohen-Macaulay vertex splittable ideals are characterized. As a consequence, we recover several Cohen-Macaulay classifications of families of monomial ideals known in the literature by new simpler combinatorial proofs.

Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. Graded semi-primary and graded $1$-absorbing primary ideals have been investigated and examined by several authors as generalizations of graded primary ideals. However, these three concepts are different. In this article, we characterize graded rings over which every graded semi-primary ideal is graded $1$-absorbing primary and graded rings over which every graded $1$-absorbing primary ideal is graded primary.

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P\bigcap S=\emptyset In this article, we introduce several results concerning graded S-prime ideals. Then we introduce the concept of graded weakly S-prime ideals which is a generalization of graded weakly prime ideals. We say that P is a graded weakly S-prime ideal of R if there exists s\in S such that for all x, y\in h(R), if 0\neq xy\in P, then sx\in P or sy\in P. We show that graded weakly S-prime ideals have many acquaintance properties to these of graded weakly prime ideals.

Let R be a commutative ring and m, n be positive integers. We define a proper submodule N of an R-module M to be (m,n)-closed if for r∈R and b∈M, r^{m}b∈N implies rⁿ∈(N:_{R}M) or b∈N. This class of submodules lies properly between the classes of prime and primary submodules. Many characterizations, properties and supporting examples concerning this class of submodules are provided. The notion of (m,n)-modules is introduced and characterized. Furhermore, the (m,n)-closed avoidance theorem is proved. Finally, the (m,n)-closed submodules in amalgamated modules are studied.

In the study of algebraic structures related to logical systems, Ideals and filters have different meanings and they are algebraic notions related to logical provable formulas. Unlike the classical Boolean lattice theory, ideals and filters are not dual notions in residuated lattices. An interesting subclass of residuated lattices is the class of triangle algebras, which is an equational representation of interval-valued residuated lattices that provides an algebraic framework for using closed intervals as truth values in fuzzy logic. The main aim of this article is to introduce and study the concept of ideals in triangle algebras and investigate the connection between ideals and filters. We first point out that the construction procedure for the filter generated by a subset of a triangle algebra established by Zahiri et al. is incorrect, and we proceed to give an alternative characterization.