Module theory MOC

Graded module

Let 𝑅 be a 𝔄-graded ring. A graded module 𝑀 over 𝑅 is a module over 𝑅 with submodules (𝑀𝛼)𝛼∈𝔄 such that 𝑀 is the direct sum ∐𝛼∈𝔄𝑀𝛼 and module

𝑅𝛼⋅𝑀𝛽⊆𝑉𝛼+𝛽

for 𝛼,𝛽 ∈𝔄.

Category of graded modules

Many of our typical module constructions carry over. See Category of graded modules.

• Graded module homomorphism
• Graded submodule, Quotient graded module
• Direct sum of graded modules

Properties

• If 𝑀 is a graded module over a graded associative or Lie algebra 𝐴, then 𝑎 ∈𝐴𝛼 acts as a ^homogenous operator of degree 𝛼.¹

• The grades of a graded module may be shifted arbitrarily, see Shifted graded module.

Further remarks

• For ℤ-gradings we adopt the physical terminology of Vacuum space for the subspace annihilated by positive grades.

See also

• Graded structure

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Footnotes

1. For the Lie case, this abuse of terminology works since The universal enveloping algebra of a Lie algebra inherits its graded structure. ↩