Dimension theory

[jjaassoonn]

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Some results are

• The ring Krull dimension and the topological dimension of the prime spectrum of a ring are the same
• A field is zero-dimensional and a PID that is not a field is one dimensional
• An Artinian ring is zero-dimensional
• The Krull dimension of ring $R$ is equal to the supremum of heights of maximal ideals 00KG
• two definitions of module length agree
• A module is finite length iff both artinian and noetherian
• length of module $M$ is equal to the sum of length of $N$ and length of $M / N$ where $N$ is a submodule of $M$.
• Noetherian ring has only finitely many minimal primes #9088
• In a zero dimensional ring, prime ideals are maximal
• Artinian rings has finitely many maximal ideals #9087
• zero-dimensional rings with finitely many prime ideals are products of its localizations: $R \cong \prod_{\mathfrak{p}}R_{\mathfrak{p}}$
• Let $M_0 \le ... \le M_n$, then $l(M_n/ M_0) = l(M_1/M_0) + l(M_2/M_1) + ... + l(M_n/M_{n-1})$ where $l$ denotes module length.
• If $f : R \to S$ is a ring homomorphism and $M$ an $S$-module, then $l_R(M) \le l_S(M)$, when $f$ is surjective then they are equal. (Note that this is expressed using [Algebra R S] and RestrictScalars R S M, instead of a literal ring hom)
• Artinian rings are noetherian ring of dimension 0.
• For 0-dimensional local ring, its maximal ideal is locally nilpotent
• In notherian ring, $I \le \sqrt{J}$ implies $I ^ n \le J$ for some $n$

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• [x] depends on: #6309