Editing Derivation (differential algebra) (section)

==Properties==

If ''A'' is a ''K''-algebra, for ''K'' a ring, and {{math|''D'': ''A'' → ''A''}} is a ''K''-derivation, then
* If ''A'' has a unit 1, then ''D''(1) = ''D''(1<sup>2</sup>) = 2''D''(1), so that ''D''(1) = 0. Thus by ''K''-linearity, ''D''(''k'') = 0 for all {{math|''k'' ∈ ''K''}}.
* If ''A'' is commutative, ''D''(''x''<sup>2</sup>) = ''xD''(''x'') + ''D''(''x'')''x'' = 2''xD''(''x''), and ''D''(''x''<sup>''n''</sup>) = ''nx''<sup>''n''−1</sup>''D''(''x''), by the Leibniz rule.
* More generally, for any {{math|''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''n''</sub> ∈ ''A''}}, it follows by [[mathematical induction|induction]] that 
*: <math>D(x_1x_2\cdots x_n) = \sum_i x_1\cdots x_{i-1}D(x_i)x_{i+1}\cdots x_n  </math>
: which is <math display="inline">\sum_i D(x_i)\prod_{j\neq i}x_j</math> if for all {{mvar|i}}, {{math|''D''(''x<sub>i</sub>'')}} commutes with <math>x_1,x_2,\ldots, x_{i-1}</math>.
* For ''n'' > 1, ''D''<sup>''n''</sup> is not a derivation, instead satisfying a higher-order Leibniz rule:
:: <math>D^n(uv) = \sum_{k=0}^n \binom{n}{k} \cdot D^{n-k}(u)\cdot  D^k(v).</math>
: Moreover, if ''M'' is an ''A''-bimodule, write
:: <math> \operatorname{Der}_K(A,M)</math>
:for the set of ''K''-derivations from ''A'' to ''M''.
* {{nowrap|Der<sub>''K''</sub>(''A'', ''M'')}} is a [[module (mathematics)|module]] over ''K''.  
* Der<sub>''K''</sub>(''A'') is a [[Lie algebra]] with Lie bracket defined by the [[commutator]]:
:: <math>[D_1,D_2] = D_1\circ D_2 - D_2\circ D_1.</math>
: since it is readily verified that the commutator of two derivations is again a derivation.
* There is an ''A''-module {{math|Ω<sub>''A''/''K''</sub>}} (called the [[Kähler differentials]]) with a ''K''-derivation {{math|''d'': ''A'' → Ω<sub>''A''/''K''</sub>}} through which any derivation {{math|''D'': ''A'' → ''M''}} factors. That is, for any derivation ''D'' there is a ''A''-module map {{mvar|φ}} with
:: <math> D: A\stackrel{d}{\longrightarrow} \Omega_{A/K}\stackrel{\varphi}{\longrightarrow} M </math>
: The correspondence <math> D\leftrightarrow \varphi</math> is an isomorphism of ''A''-modules:
:: <math> \operatorname{Der}_K(A,M)\simeq \operatorname{Hom}_{A}(\Omega_{A/K},M)</math>
* If {{math|''k'' ⊂ ''K''}} is a [[subring]], then ''A'' inherits a ''k''-algebra structure, so there is an inclusion
:: <math>\operatorname{Der}_K(A,M)\subset \operatorname{Der}_k(A,M) ,</math>
: since any ''K''-derivation is ''a fortiori'' a ''k''-derivation.