[[Ring theory MOC]]
# Algebra over a commutative ring

An **algebra** $V$ over a [[commutative ring]] $R$ is an $R$-[[module]] $V$ equipped with a [[Multilinear map|bilinear]] product $(\cdot): V \times V \to V$, #m/def/falg
i.e. for any $x,y,z \in V$ and $\alpha,\beta \in R$

1. $(x+y)z = xz + yz$ ^A1
2. $z(x+y) = zx + zy$ ^A2
3. $(\alpha x)(\beta y)=(\alpha  \beta)(xy)$ ^A3

Thus it is a [[Magma object]] in [[Category of modules over a commutative ring|$\lMod{R}$]].

## Examples

- [[Endomorphism ring]]

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