Krull Dimension of A Module
If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module. That is, we define it by the formula:
where, the annihilator, is the kernel of the natural map of R into the ring of -linear endomorphisms on .
In the language of schemes, finite type modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.
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