Definition:Unitary Module

Definition
Let $\left({R, +_R, \times_R}\right)$ be a ring.

Let $\left({G, +_G}\right)$ be an abelian group. Let $\left({G, +_G, \circ}\right)_R$ be a module over $R$ such that the ring $\left({R, +_R, \times_R}\right)$ is a ring with unity whose unity is $1_R$.

Then $\left({G, +_G, \circ}\right)_R$ is a unitary module over $R$ or unitary $R$-module iff:
 * $(4): \quad \forall x \in G: 1_R \circ x = x$.