nLab supercommutative ring

Idea

A supercommutative ring is an $\mathbb{Z}$-supercommutative algebra.

Definition

A supercommutative ring is a super ring $R$, such that

• for all $a:R$, and $b:R$, $\mathcal{D}_0(a) \cdot \mathcal{D}_0(b) = \mathcal{D}_0(b) \cdot \mathcal{D}_0(a)$
• for all $a:R$, and $b:R$, $\mathcal{D}_1(a) \cdot \mathcal{D}_0(b) = \mathcal{D}_0(b) \cdot \mathcal{D}_1(a)$
• for all $a:R$, and $b:R$, $\mathcal{D}_0(a) \cdot \mathcal{D}_1(b) = \mathcal{D}_1(b) \cdot \mathcal{D}_0(a)$
• for all $a:R$, and $b:R$, $\mathcal{D}_1(a) \cdot \mathcal{D}_1(b) = - \mathcal{D}_1(b) \cdot \mathcal{D}_1(a)$

See also

• ring

• differential ring

• exterior ring

• graded ring

• super abelian group

• supercommutative algebra

• super ring