Module theory MOC

Graded module

Let be a -graded ring. A graded module over is a module over with submodules such that is the direct sum and module

for .

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 is a graded module over a graded associative or Lie algebra , then acts as a ^homogenous operator of degree .¹

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

Further remarks

• For -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. ↩