# Definition:Algebra over Ring

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## Definition

Let $R$ be a commutative ring.

An **algebra over $R$** is an ordered pair $\left({A, *}\right)$ where:

- $A$ is an $R$-module
- $*: A^2 \to A$ is an $R$-bilinear mapping

## Commutativity of the Ring

Because the definition of bilinear mapping is intricate in the case of noncommutative rings, it is important that $R$ be commutative. It does not necessarily have to be a ring with unity.

## Also defined as

Sources who only deal with rings with unity often define an **algebra** as one whose underlying module is unital.

Especially in commutative algebra, an **algebra** over a commutative ring with unity $R$ is often defined as a **unital associative commutative algebra**.

## Also see

It can be considered to be a generalization of an algebra over a field in which:

- the vector space is replaced by a module
- the field is replaced by a commutative ring.

- Results about
**algebras**can be found here.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**algebra**:**6.**