A commutative group $(M,+)$ graded over $\mathbb{Z}/2\mathbb{Z}$ is a $\mathbb{K}$ -module if it is equipped with a bilinear application $M\times\mathbb{K}\rightarrow M$ such that, for any $\alpha,\beta\in\mathbb{K}$ and $m,n\in M$ we have

$\displaystyle(m\alpha)\beta=m(\alpha\beta),\ \ \ p(m\alpha)=p(m)+p(\alpha).$

We denote by $M_{0}$ and $M_{1}$ the $\mathbb{K}_{0}$ -submodules composed of even and odd elements.

We say that $A$ is a $\mathbb{K}$ -superalgebra if it a $\mathbb{K}$ -module equipped of a distributive application $A\times A\stackrel{{\scriptstyle\cdot}}{{\rightarrow}}A$ such that

$\displaystyle p(a\cdot b)=p(a)+p(b),\ \ \ \ (a\cdot b)\alpha=a\cdot(b\alpha)=(-1)^{p(b)p(\alpha)}(a\alpha)\cdot b$

for any $a,b\in A$ and $\alpha\in\mathbb{K}$ . We say that $A$ is commutative if $a\cdot b=(-1)^{p(a)p(b)}b\cdot a$ for $a,b\in A$ , and $c^{2}=0$ for $c\in A_{1}$ .