Graded module

Let $𝑅$ be a $𝔄$ -graded ring. A graded module $𝑀$ over $𝑅$ is a module over $𝑅$ with submodules $(𝑀 𝛼) 𝛼 \in 𝔄$ such that $𝑀$ is the direct sum $∐ 𝛼 \in 𝔄 𝑀 𝛼$ and #m/def/module

𝑅 𝛼 \cdot 𝑀 𝛽 \subseteq 𝑉 𝛼 + 𝛽

for $𝛼, 𝛽 \in 𝔄$ .

Category of graded modules

Many of our typical module constructions carry over. See $𝖦 𝗋 𝔄 𝑅 𝖬 𝗈 𝖽$ .

Properties

If $𝑀$ is a graded module over a graded associative or Lie algebra $𝐴$ , then $𝑎 \in 𝐴 𝛼$ 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.

Graded module

Category of graded modules

Properties

Further remarks

See also

Footnotes