Chain ring

In commutative algebra , a ring is called a chain ring or a catenary ring if non-refinable prime ideal chains of two nested prime ideals always have the same length. Catenary rings have simple dimensional theoretical properties.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

definition

If a ring is, then a prime ideal chain is a sequence of prime ideals ( ): ${\displaystyle R}$ ${\displaystyle p_{i}\subset R}$

${\displaystyle p_{0}\subsetneq p_{1}\dots \subsetneq p_{n}}$

The length of this prime ideal chain is . Such a prime ideal chain is called a chain that can no longer be refined if there is no prime ideal such that ${\displaystyle n}$ ${\displaystyle p_{i}\neq q\subset R}$

${\displaystyle p_{0}\subsetneq \dots \subsetneq q\subsetneq \dots \subsetneq p_{n}}$

is a prime ideal chain.

If there is a ring, it is called a catenary or a chain ring when it applies to all prime ideals that all prime ideal chains that cannot be refined, which begin with and end with , have the same length. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle p\subsetneq q\subset R}$ ${\displaystyle p}$ ${\displaystyle q}$

properties

• If a Noetherian ring is catenary, then also every residual class ring and every localization .
• Catenary is a local property : A Noetherian ring is catenary if and only if the ring is catenary for every maximal ideal . ${\displaystyle R}$ ${\displaystyle m\subset R}$ ${\displaystyle R_{m}}$
• If it is Noetherian, catenary and zero divisor and also all maximum ideals have the same height (e.g. , see below), then every residual class ring according to a prime ideal of has this property. For every prime ideal then applies: ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle K[X_{1},\dots ,X_{n}]}$ ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle p\subset A}$

${\displaystyle \mathrm {ht} (p)+\mathrm {dim} (A/p)=\mathrm {dim} A}$.

Examples

• If there is a body, the ring is catalytic. ${\displaystyle K}$ ${\displaystyle K[X_{1},\dotsc X_{n}]}$
• Every Cohen-Macaulay ring , especially every regular ring , is catenary.

literature

• H. Matsumura, Commutative algebra , 1980, ISBN 0-8053-7026-9 .
• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411