Remark

(1) The notion of derivation can be defined for non-graded modules by agreeing to give these modules the trivial graduation.

(2) If only $\varepsilon$-derivations of given degree $\delta$ are considered, the commutation factor $\varepsilon$ may be disposed of as follows: the bilinear mapping $\lambda_2 : A \times B' \to B''$ is modified by replacing it by the bilinear mapping $\lambda'_2 : A \times B' \to B''$ such that, for every homogeneous $a$ in $A$ and all $b' \in B'$,

Then $d$ is a derivation relative to the bilinear mappings $\mu, \lambda_1, \lambda'_2$.

The general definition of $\varepsilon$-derivations given above is especially used in two cases:

Case (I): $A = B, A' = B', A'' = B''$ and the three bilinear mappings $\mu, \lambda_1, \lambda_2$ are equal to the same mapping.

Case (II): $A = A' = A'', B = B' = B''$, so that (for $\mu$) $A$ is a graded algebra and the two K-bilinear mappings.

are such that the corresponding K-linear mappings $B \otimes_K A \to B, A \otimes_K B \to B$ are graded of degree 0. An $\varepsilon$-derivation of degree $\delta$ of $A$ into $B$ is then a graded K-linear mapping $d : A \to B$ of degree $\delta$, such that for every homogeneous $x$ in $A$ and every $y \in A$, we have the relation

Consider in particular in case (II) the case where A is a unital associative K-algebra and $\lambda_1$ and $\lambda_2$ are the external laws of an (A, A)-bimodule (§ 4, no. 3, Definition 3). This holds notably when A and B are two unital associative K-algebras, a unital homomorphism of graded K-algebras $\rho : A \to B$ is given and an (A, A)-bimodule structure is considered on B defined by the two external laws

for $a \in A, b \in B$.

Cases (I) and (II) have the following case in common: consider a graded K-algebra A, take $B = A$, mappings (7) both being multiplication on A. We then speak of an $\varepsilon$-derivation (or (K, $\varepsilon$)-derivation) of the graded algebra A: it is a graded K-linear mapping of A into itself, of degree $\delta$, satisfying (8) for every homogeneous $x$ in A and all $y \in A$. In particular if A is a graded ring, considered as an (associative) $\mathbf{Z}$-algebra, we speak of the $\varepsilon$-derivation of the ring A.

Let A be a unital commutative associative K-algebra and B an A-module; when we speak of a derivation of A into B, it will always be understood that we mean with the A-bimodule structure on B derived from its A-module structure; then the formula

holds for such a derivation $d : A \to B$.