Dedekindring

A Dedekind ring (after Richard Dedekind , also Dedekind area or ZPI ring ) is a generalization of the ring of whole numbers . The applications of this term can be found mainly in the mathematical sub-areas of algebraic number theory and commutative algebra , especially in ideal theory .

definition

A Dedekind ring is at most a one-dimensional , Noetherian , normal integrity ring .

Some authors claim that Dedekind rings are one-dimensional, which means that bodies are not Dedekind rings by definition. However, this is not common.

properties

• Analogous to the unique decomposition of integers into prime numbers applies to Dedekind domain that in them every ideal a unique Z nstallation in P rim i deale has. Dedekind rings are precisely those integrity rings that are ZPI rings .

• Zero-dimensional Dedekind rings are bodies.

• One-dimensional local Dedekind rings are precisely the discrete evaluation rings .

• Every broken ideal different from the zero ideal can be inverted via a Dedekind ring.

• Factorial Dedekind rings are main ideal rings . Conversely, every main ideal ring is a factorial Dedekind ring.

Examples

• Every main ideal ring (and therefore every discrete evaluation ring ) is a Dedekind ring .
• If a main ideal ring is a finite extension of its quotient field , the whole ending of in is a Dedekind ring . This applies in particular to whole rings in number fields , for example ${\displaystyle A}$ ${\displaystyle L}$ ${\displaystyle A}$ ${\displaystyle L}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}].}$
• Localizations of Dedekind rings are again Dedekind rings.

No Dedekind rings are:

• ${\displaystyle \mathbb {Z} [X]}$ (two-dimensional),
• ${\displaystyle \mathbb {Z} [{\sqrt {5}}]}$ (not normal),
• ${\displaystyle \mathbb {Z} [X]/(X^{2})}$and (no integrity rings), ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$
• the ring of algebraic integers, ie the whole closure of in an algebraic closure of the rational numbers (not noetherian). ${\displaystyle \mathbb {Z} }$ ${\displaystyle {\overline {\mathbb {Q} }}}$

literature

• LA Bokut ': Dedekind ring . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).