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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]].\nA *commutative $R$-algebra* is a pair $(A, f)$ where $A$ is a commutative ring and $f \\colon R \\to A$ is a [[ring homomorphism|0GGE0Z0JQ1]].\nThe map $f$ is the *structure map* of the algebra.\nFor $r \\in R$ and $a \\in A$ one writes $r a = f(r) a$.\nIf $(A, f)$ and $(B, g)$ are commutative $R$-algebras, an *$R$-algebra homomorphism* is a ring homomorphism $h \\colon A \\to B$ with $h \\circ f = g$.\n\n## Remarks\n\nExample: for $R = \\Z$ and $A = \\Z/n\\Z$, the reduction map $\\Z \\to \\Z/n\\Z$ is a ring homomorphism, so $\\Z/n\\Z$ is a $\\Z$-algebra.\nExample: the polynomial ring $R[x_1, \\ldots, x_n]$ with the inclusion of constants is an $R$-algebra.\nNon-example: the ring $M_2(\\R)$ of $2 \\times 2$ real matrices receives the ring homomorphism $c \\mapsto c I$ from $\\R$.\nBut $M_2(\\R)$ is not commutative, so it is an associative $\\R$-algebra and not a commutative one.\nEquivalent form: a commutative $R$-algebra is a commutative ring $A$ with an $R$-[[module|09VDS2W1E8]] structure such that $r(ab) = (ra)b$ for all $r \\in R$ and $a, b \\in A$; the structure map is then $r \\mapsto r \\cdot 1$.\nThe notion of a [[finite type algebra|AC5TFFQT9M]] and the [[tensor product of algebras|1735VGRT2P]] are built on this definition.\n",
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