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 #m/def/module

for .

Category of graded modules

Many of our typical module constructions carry over. See .

• 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

#state/tidy | #lang/en | #SemBr