# 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.

## 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).${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle {\ overline {\ mathbb {Q}}}}$ 