A note on uniformly pr-submodules

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. © Institute of Mathematics, Czech Academy of Sciences 2026.