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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $A$ be a commutative [[$R$-algebra|DPBJ04JMH8]].\nFor an $A$-[[module|0NZ8JA25WK]] $M$, an *$R$-derivation* $D \\colon A \\to M$ is an $R$-linear map with $D(ab) = a \\, D(b) + b \\, D(a)$ for all $a, b \\in A$; see [[derivation|3J89REV9WQ]].\nThe *module of Kähler differentials* $\\Omega_{A/R}$ is the $A$-module with one generator $da$ for each $a \\in A$ and the following relations, for all $a, b \\in A$ and $r \\in R$:\n\n1. $d(a + b) = da + db$.\n2. $d(ab) = a \\, db + b \\, da$.\n3. $d(r \\cdot 1) = 0$.\n\nThe map $d \\colon A \\to \\Omega_{A/R}$, $a \\mapsto da$, is the *universal derivation*.\n\n## Remarks\n\nThe map $d$ is an $R$-derivation, and it is universal: for each $R$-derivation $D \\colon A \\to M$ there is a unique $A$-linear map $\\varphi \\colon \\Omega_{A/R} \\to M$ with $D = \\varphi \\circ d$.\nAn equivalent form: if $I$ is the kernel of the multiplication map $A \\otimes_R A \\to A$ (see [[tensor product of algebras|1735VGRT2P]]), then $\\Omega_{A/R} \\cong I/I^2$, with $da$ corresponding to the class of $a \\otimes 1 - 1 \\otimes a$.\n\nExample: for the [[polynomial ring|0SDC934R52]] $A = R[x_1, \\ldots, x_n]$, the module $\\Omega_{A/R}$ is free with basis $dx_1, \\ldots, dx_n$, and $df = \\sum_i \\frac{\\partial f}{\\partial x_i} \\, dx_i$, with [[formal partial derivatives|0EGN1E1JPB]].\nFor a quotient $A = R[x]/(f)$, the module $\\Omega_{A/R}$ is $A \\, dx / (f'(x) \\, dx) \\cong A/(f'(x))$.\nSo $\\Omega_{\\Q(\\sqrt 2)/\\Q} = 0$, because $f = x^2 - 2$ and $f'(\\sqrt 2) = 2\\sqrt 2$ is a unit.\nFor $A = \\F_p(t^{1/p})$ over $R = \\F_p(t)$, with $f = x^p - t$, one has $f' = 0$, so $\\Omega_{A/R} \\cong A \\ne 0$.\n\nNon-example: the identity map $A \\to A$ is not a derivation when $A \\ne 0$, because it gives $1 = D(1 \\cdot 1) = 2 \\cdot D(1) = 2$, which fails in $A$ unless $1 = 0$.\n",
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