Commutative algebra over a ring

Summary

A commutative algebra over a commutative ring $R$ is a commutative ring $A$ together with a ring homomorphism $R \to A$. The homomorphism lets one multiply elements of $A$ by elements of $R$, so $A$ is at the same time a ring and an $R$-module. Most rings in commutative algebra and algebraic geometry come with such a structure, for example polynomial rings over $R$ and their quotients. The notion lets one compare rings over a fixed base and speak of $R$-algebra homomorphisms.