Let R be a commutative ring with nonzero identity, and let M be a nonzero unital R-module. In this article, we introduce the concept of 2-absorbing quasi primary submodules which is a generalization of prime submodules. We define 2-absorbing quasi primary submodule as a proper submodule N of M having the property that abm ∈ N, then ab ∈ (N : R M) or am ∈ rad M (N) or bm ∈ rad M (N). Various results and examples concerning 2-absorbing quasi primary submodules are given.

Abstract:In this paper we investigate graded compactly packed rings, which is de ned as; if any graded ideal I of R is contained in the union of a family of graded prime ideals of R, then I is actually contained in one of the graded prime ideals of the family. We give some characterizations of graded compactly packed rings. Further, we examine this property on h − Spec(R). We also de ne a generalization of graded compactly packed rings, the graded coprimely packed rings. We show that R is a graded compactly packed ring if and only if R is a graded coprimely packed ring whenever R be a graded integral domain and h − dim R = .

In this study, Catalan transformation 𝐶𝑆 𝑘,𝑛 of 𝑘−Jacobsthal-Lucas sequences 𝑆 𝑘,𝑛 is defined. In addition, the transformation of C𝑆 𝑘,𝑛 is written as the product of the Catalan matrix C which is the lower triangular matrix and the 𝑆 𝑘 matrix of type 𝑛 𝑥 1, and the Hankel transformations of some 𝑘−Jacobsthal-Lucas numbers is found.

Let R be a commutative ring with nonzero identity and I a proper ideal of R. Then I is called a uniformly pr-ideal if there exists N ∈ N such that ab ∈ I with ann(a) = 0 then b N ∈ I. We say that the smallest N ∈ N is called order of I and denoted by ordR(I) = N. In this paper, we give some examples and characterizations of this new class of ideals.