Let $A$ be a $K$ -superalgebra in a variety $\mathfrak{V}$ (later $\mathfrak{V}$ will be the variety of Jordan or Lie superalgebras). Let ${\cal{U}}$ be a $K$ -supermodule equipped with a pair of $K$ -bilinear mappings $(a,u)\mapsto au,\,(a,u)\mapsto ua,\,a\in A,\,u\in{\cal{U}}$ , of $A\times{\cal{U}}$ into ${\cal{U}}$ of degree 0. Then $X=A\oplus{\cal{U}}$ is a $K$ -supermodule on which we define a multiplication by

$\displaystyle(a+u)(b+v)=ab+av+ub$

(1.5)

for all $a,b\in A,\,u,v\in{\cal{U}}$ . Since this product is $K$ -bilinear, $X$ is a superalgebra over $K$ , called the split null extension of ${\cal{A}}$ determined by the bilinear mappings of $A$ and ${\cal{U}}$ (see [ J , Chap. II, Sect., 5] for the classical case). If $X$ is a superalgebra in the variety $\mathfrak{V}$ then we say that ${\cal{U}}$ is a $\mathfrak{V}$ -module for $A$ . In this case, for each $\alpha\in\mathbb{Z}_{2}$ let $(\operatorname{Der}_{K}(A,{\cal{U}}))_{\alpha}$ be the space of all homogeneous $K$ -module homomorphisms $d\in\operatorname{Hom}_{K}(A,{\cal{U}})_{\alpha}$ satisfying for all homogeneous $x\in A$ and all $y\in A$

$$d(xy)=d(x)y+(-1)^{\alpha|x|}xd(y).$$

We define

$$\operatorname{Der}_{K}(A,{\cal{U}})=(\operatorname{Der}_{K}(A,{\cal{U}}))_{\operatorname{\bar{0}}}\oplus(\operatorname{Der}_{K}(A,{\cal{U}}))_{\operatorname{\bar{1}}}.$$

Then it is easy to see that $\operatorname{Der}_{K}(A,{\cal{U}})$ is a submodule of $\operatorname{End}_{K}{\cal{M}}$ and hence a $K$ -supermodule. The elements of $\operatorname{Der}_{K}(A,{\cal{U}})$ are called the $K$ -derivations (from $A$ to ${\cal{U}}$ ) . If $A={\cal{U}}$ , we simply write $\operatorname{Der}_{K}A$ .