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a graded submodule of M if and only if it is a submodule of M and satisfies . The kernel and the image of a morphism of graded modules are graded submodules.

Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring.

Given a graded module , the -twist of is

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