We introduce the notion of S-1-absorbing primary submodule as an extension of 1-absorbing primary submodules. Let S be a multiplicatively closed subset of a ring R and M be an R-module. A submodule N of M with (N:RM) ∩ S = ∅ is said to be S-1-absorbing primary if there exists s ∈ S such that whenever abm ∈ N for some non-units a,b ∈ R and m ∈ M, then either sab ∈ (N:RM) or sm ∈ M-rad(N). We examine several properties of this concept and provide some characterizations. In addition, S-1-absorbing primary avoidance theorem and S-1-absorbing primary property for idealization and amalgamation are presented.
Let M be a nonzero unital module over a commutative ring R with a nonzero identity. A proper submodule N of M is said to be a uniformly pr-submodule if there exists a fixed n ∈ ℕ such that whenever am ∈ N with ann(m) = 0 for some a ∈ R and m ∈ M, then an ∈ (N: M). Indeed, uniformly pr-submodules are the extension of uniformly pr-ideals to modules in the sense of R. N. Uregen (2019). In addition to giving fundamental properties of uniformly pr-submodules, we also examine the relations between uniformly pr-submodules and other classical submodules such as uniformly primary submodules, r-submodules, sr-submodules and pr-submodules. Furthermore, we characterize some important classes of rings/modules such as total quotient rings and simple modules in terms of our new concept.
