# Definition:Ring of Linear Operators

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## Definition

Let $R$ be a ring.

Let $G$ be an $R$-module.

Let $\map {\LL_R} G$ denote the set of all linear operators on $G$.

Let $+$ and $\circ$ be the binary operations on $\map {\LL_R} G$ defined such that:

- $+$ denotes pointwise addition
- $\circ$ denotes composition of linear operators.

Then the algebraic structure:

- $\struct {\map {\LL_R} G, +, \circ}$ is a ring

is known as the **ring of linear operators on $G$**.