nLab super ring

Idea

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

Definition

A super ring is a ring $R$ with decomposition functions $\mathcal{D}_0:R \to R$ and $\mathcal{D}_1:R \to R$, such that

• for all $a:R$, $a = \mathcal{D}_0(a) + \mathcal{D}_1(a)$
• for all $a:R$, and $b:R$, $\mathcal{D}_0(a + b) = \mathcal{D}_0(a) + \mathcal{D}_0(b)$
• for all $a:R$, and $b:R$, $\mathcal{D}_1(a + b) = \mathcal{D}_1(a) + \mathcal{D}_1(b)$
• for all $a:R$, and $b:R$, $\mathcal{D}_0(a \cdot b) = \mathcal{D}_0(a) \cdot \mathcal{D}_0(b) - \mathcal{D}_1(a) \cdot \mathcal{D}_1(b)$
• for all $a:R$, and $b:R$, $\mathcal{D}_1(a \cdot b) = \mathcal{D}_0(a) \cdot \mathcal{D}_1(b) + \mathcal{D}_1(a) \cdot \mathcal{D}_0(b)$
• for all $a:R$, and $b:R$, $\mathcal{D}_0(1) = 1$
• for all $a:R$, and $b:R$, $\mathcal{D}_1(1) = 1$
• for all $a:R$, $\mathcal{D}_0(\mathcal{D}_0(a)) = \mathcal{D}_0(a)$
• for all $a:R$, $\mathcal{D}_0(\mathcal{D}_1(a)) = 0$
• for all $a:R$, $\mathcal{D}_1(\mathcal{D}_0(a)) = 0$
• for all $a:R$, $\mathcal{D}_1(\mathcal{D}_1(a)) = \mathcal{D}_1(a)$

As a result, the image of the two decompostion functions $\im(\mathcal{D}_0)$ and $\im(\mathcal{D}_1)$ are rings and there exists an abelian group isomorphism $i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1)$, where $F:Ring \to Ab$ is a forgetful functor and $F(A) \otimes F(B)$ is the tensor product of abelian groups.

The elements of $\im(\mathcal{D}_0)$ are called even elements or bosonic elements, and the elements of $\im(\mathcal{D}_1)$ are called odd elements or fermionic elements.

See also

• ring

• differential ring

• exterior ring

• Clifford ring

• graded ring

• super abelian group

• super algebra

• supercommutative ring