On Weakly (1, n)-Submodules and Weakly n-Submodules of Modules Over Commutative Rings

Let R be a commutative ring with a nonzero identity and M a nonzero unital R-module. We introduce the concepts of weakly n-submodules and weakly (1, n)-submodules as module-theoretic generalisations of the weakly n-ideal and weakly (1, n)-ideal. A proper submodule N of M is called a weakly n-submoduleif whenever 0 ̸= am ∈N for some a ∈R and m ∈M , then a ∈(N :R M )or m ∈Nil(M )·M , where Nil(M ) = annR(M ). Similarly, N is called a weakly (1, n)-submodule if whenever 0 ̸= abm ∈N for some nonunit elementsa, b ∈R and m ∈M , then ab ∈(N :R M ) or m ∈Nil(M )·M . Every weaklyn-submodule is a weakly (1, n)-submodule, and every weakly (1, n)-submoduleis weakly 1-absorbing primary. We provide a six-fold characterisation, provestructure theorems classifying the rings and modules over which every propersubmodule belongs to these classes in particular, we show that for faithfulnitely generated multiplication modules, every proper submodule is weakly (1, n) if and only if R is a UN-ring or a product of two elds and investigatebehaviour under homomorphisms, localisations, and quotient modules.