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  "body": "\n## Summary\n\nA commutative algebra over a commutative ring $R$ is a [[commutative ring|0GYGHKX834]] $A$ together with a [[ring homomorphism|0GGE0Z0JQ1]] $R \\to A$.\nThe homomorphism lets one multiply elements of $A$ by elements of $R$, so $A$ is at the same time a ring and an $R$-module.\nMost rings in commutative algebra and algebraic geometry come with such a structure, for example polynomial rings over $R$ and their quotients.\nThe notion lets one compare rings over a fixed base and speak of $R$-algebra homomorphisms.\n\n## Explanations\n\n### L1 human\n\nAn $R$-algebra is a ring whose elements can be scaled by the elements of a base ring $R$.\nThink of the polynomial ring $\\R[x]$: one can add and multiply polynomials, and one can also multiply a polynomial by a real number.\nThe map that sends a real number $c$ to the constant polynomial $c$ is the structure map $\\R \\to \\R[x]$.\nEvery commutative ring is an algebra over $\\Z$ in exactly one way, since there is exactly one ring homomorphism $\\Z \\to A$.\n",
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