Let R be a commutative ring with 1 = 0. In this paper, we introduce the concept of 2-absorbing primary ideal which is a generalization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

Abstract. Let R be a commutative ring with 1 = 0. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, c ∈ R and 0 = abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.

In this article, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each m ∈ M and elements a, b ∈ R, abm ∈ N implies that am ∈ N or bm ∈ N . We introduce the concept of "classical 2-absorbing submodules" as a generalization of "classical prime submodules". We say that a proper submodule N of M is a classical 2-absorbing submodule if whenever a, b, c ∈ R and m ∈ M with abcm ∈ N , then abm ∈ N or acm ∈ N or bcm ∈ N .

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A new kind of fuzzy module over a fuzzy ring is introduced by generalizing Yuan and Lee's definition of the fuzzy group and Aktaş and Çağman's definition of fuzzy ring. The concepts of fuzzy submodule, and fuzzy module homomorphism are studied and some of their basic properties are presented analogous of ordinary module theory.