Algebra theory MOC

Graded algebra

Let (𝑀, +,0) be a monoid. An algebra (𝐴, ⋅) over 𝕂 is said to be 𝑀-graded iff it is an 𝑀-graded vector space 𝐴 =∐𝛼∈𝑀𝐴𝛼 such that falg

𝐴𝛼⋅𝐴𝛽⊆𝐴𝛼+𝛽

for any 𝛼,𝛽 ∈𝑀. If (𝐴, ⋅) is a K-monoid, this definition is equivalent to that of a graded ring, and hence 1 ∈𝐴 +0.

Category of graded algebras

Many of our typical algebra constructions carry over. These motivate Category of graded algebras.

• Graded algebra homomorphism
• Graded subalgebra, Quotient graded algebra

Properties

• If 𝑀 ≤𝕂+, the degree operator on 𝐴 is a derivation.
• If 𝐴 is commutative, then 𝑀 must be abelian/

Examples

• Tensor algebra

See also

• Graded structure

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