# nLab supercommutative ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

supersymmetry

# Contents

## 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 $b:R$, $\mathcal{D}_1(a) \cdot \mathcal{D}_1(b) = - \mathcal{D}_1(b) \cdot \mathcal{D}_1(a)$