Definition:Module of Homomorphisms Between Modules
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Definition
Let $R$ be a commutative ring.
Let $M$ and $N$ be $R$-modules.
Let $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ denote the set of $R$-module homomorphisms from $M$ to $N$.
Let:
- $+$ be the operation on $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ defined by $f + g: m \mapsto \map f m + \map g m$
- $\circ$ be defined as $\lambda \circ f: m \mapsto \lambda \map f m$
Then $\struct {\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}, + , \circ}$ is called the module of homomorphisms between $M$ and $N$.