# Definition:Algebra Defined by Ring Homomorphism

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## Definition

Let $R$ be a commutative ring.

Let $\struct {S, +, *}$ be a ring.

Let $f : R \to S$ be a ring homomorphism.

Let the image of $f$ be a subset of the center of $S$.

Let $S_R$ be the module defined by the ring homomorphism $f$.

The **algebra defined by $f$** is the algebra over $R$ defined as:

- $\struct {S_R, *}$

## Also see

### Properties

- Algebra Defined by Ring Homomorphism is Associative
- Algebra Defined by Ring Homomorphism on Ring with Unity is Unitary
- Algebra Defined by Ring Homomorphism on Commutative Ring is Commutative

## Sources

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