supercommutative ring (changes) in nLab

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Contents

Idea

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

Definition

A supercommutative ring is a super ring RR, such that

• for all a:Ra:R, and b:Rb:R, 𝒟 0(a)⋅𝒟 0(b)=𝒟 0(b)⋅𝒟 0(a)\mathcal{D}_0(a) \cdot \mathcal{D}_0(b) = \mathcal{D}_0(b) \cdot \mathcal{D}_0(a)
• for all a:Ra:R, and b:Rb:R, 𝒟 1(a)⋅𝒟 0(b)=𝒟 0(b)⋅𝒟 1(a)\mathcal{D}_1(a) \cdot \mathcal{D}_0(b) = \mathcal{D}_0(b) \cdot \mathcal{D}_1(a)
• for all a:Ra:R, and b:Rb:R, 𝒟 0(a)⋅𝒟 1(b)=𝒟 1(b)⋅𝒟 0(a)\mathcal{D}_0(a) \cdot \mathcal{D}_1(b) = \mathcal{D}_1(b) \cdot \mathcal{D}_0(a)
• for all a:Ra:R, and b:Rb:R, 𝒟 1(a)⋅𝒟 1(b)=−𝒟 1(b)⋅𝒟 1(a)\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