Ring theory MOC

Algebra over a commutative ring

An algebra $𝑉$ over a commutative ring $K$ is an $K$ -module $𝑉$ equipped with a bilinear product $(\cdot) : 𝑉 \times 𝑉 \to 𝑉$ , #m/def/ralg i.e. for any $𝑥, 𝑦, 𝑧 \in 𝑉$ and $𝛼, 𝛽 \in K$

$(𝑥 + 𝑦) 𝑧 = 𝑥 𝑧 + 𝑦 𝑧$
$𝑧 (𝑥 + 𝑦) = 𝑧 𝑥 + 𝑧 𝑦$
$(𝛼 𝑥) (𝛽 𝑦) = (𝛼 𝛽) (𝑥 𝑦)$

Thus it is a Magma object in $K 𝖬 𝗈 𝖽$ .

Examples

Endomorphism ring

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