2015/03/01 by Mostafanasab, Hojjat, Yetkin, Ece, Tekir, Ünsal +1
#13A15 #13F05 #13G05 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1503.00308
All rings are commutative with 1≠0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a,b∈ R and m∈ M and abm∈ N, then am∈ M-rad(N) or bm∈ M-rad(N) or ab∈(N:RM). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K⊆ N for some ideals I1,I2 of R and some submodule K of M, then I1I2⊆(N:RM) or I1K⊆ M-rad(N) or I2K⊆ M-rad(N). We prove that for a submodule N of an R-module M if M-rad(N) is a prime submodule of M, then N is a 2-absorbing primary submodule of M. If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M, then (N:RM) is a 2-absorbing primary ideal of R and M-rad(N) is a 2-absorbing submodule of M.