Definition:Ring Representation

Definition

Let $R$ be a ring.

Let $M$ be an abelian group.

A ring representation of $R$ on $M$ is a ring homomorphism from $R$ to the endomorphism ring $\map {\operatorname {End} } M$.

Unital Ring Representation

Let $R$ be a ring with unity.

Let $M$ be an abelian group.

A unital ring representation of $R$ on $M$ is a ring representation $R \to \map {\operatorname {End} } M$ which is unital.

That is, it is a unital ring homomorphism from $R$ to the endomorphism ring $\map {\operatorname {End} } M$.

Also see

• Definition:Ring Antirepresentation
• Definition:Module over Ring
• Correspondence between Ring Actions and Ring Representations