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 ( ⋅) :𝑉 ×𝑉 →𝑉, falg i.e. for any 𝑥,𝑦,𝑧 ∈𝑉 and 𝛼,𝛽 ∈K

1. (𝑥 +𝑦)𝑧 =𝑥𝑧 +𝑦𝑧
2. 𝑧(𝑥 +𝑦) =𝑧𝑥 +𝑧𝑦
3. (𝛼𝑥)(𝛽𝑦) =(𝛼𝛽)(𝑥𝑦)

Thus it is a Magma object in Category of modules over a commutative ring.

Examples

• Endomorphism ring

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