On weakly 1-absorbing primary submodules

In this paper, we introduce weakly 1-absorbing primary submodules of modules over commutative rings. Let R be a commutative ring with a nonzero identity and M be a nonzero unital module. A proper submodule N of M is said to be a weakly 1-absorbing primary submodule if whenever 0 not equal abm is an element of N for some nonunit elements a, b is an element of R and m is an element of M, then ab is an element of (N : M) or m is an element of M-rad(N), where M-rad(N) is the prime radical of N. Many properties and characterizations of weakly 1-absorbing primary submodules are given. We also give the relations between weakly 1-absorbing primary submodules and other classical submodules such as weakly prime, weakly primary, weakly 2-absorbing primary submodules. Also, we use them to characterize simple modules.