Weakly sdf-Absorbing Submodules Over Commutative Rings

Let $R$ be a commutative ring with identity and $M$ a unital $R$-module. A proper submodule $N$ of $M$ is called a weakly square-difference factor absorbing submodule (briefly, weakly sdf-absorbing submodule) if for all $a,b\in R$ and $m\in M$, the condition $0\neq(a^{2}-b^{2})m\in N$ implies that either $(a+b)m\in N$ or $(a-b)m\in N$. In this paper, we investigate various characterizations and properties of weakly sdf-absorbing submodules in several module constructions. To construct new examples, we explore connections of this class with idealization rings and amalgamation modules. Examples illustrating the distinction between weakly sdf-absorbing and sdf-absorbing submodules are also provided.