a K-derivation through which any derivation factors. That is, for any derivation D there is a A-module map with The correspondence is an isomorphism of A-modules: If is a subring, then A inherits a k-algebra structure, so there is an inclusion since any K-derivation is a fortiori a k-derivation. Graded derivations Given a graded algebra A and a homogeneous linear map D of grade on A, D is a homogeneous derivation if for every homogeneous element a and every element b of A for a commutator factor . A graded derivation is sum of homogeneous derivations with the same ε. If , this definition reduces to the usual case. If , however, then for odd , and D is called an anti-derivation. Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms. Graded derivations of superalgebras (i.e. Z2-graded algebras) are often called superderivations. Related notions Hasse–Schmidt derivations are K-algebra homomorphisms Composing further with the map which sends a formal power series to the coefficient gives a derivation. See also In differential geometry derivations are tangent vectors Kähler differential Hasse derivative p-derivation Wirtinger derivatives Derivative of the exponential map References . . . . Differential algebra