Dedekind Ring

A Dedekind ring is a commutative ring in which the following hold.

1. It is a Noetherian ring and a integral domain.

2. It is the set of algebraic integers in its field of fractions.

3. Every nonzero prime ideal is also a maximal ideal. Of course, in any ring, maximal ideals are always prime.

The main example of a Dedekind domain is the ring of algebraic integers in a number field, an extension field of the rational numbers. An important consequence of the above axioms is that every ideal can be written uniquely as a product of prime ideals. This compensates for the possible failure of unique factorization of elements into irreducibles.

REFERENCES:

Atiyah, M. F. and MacDonald, I. G. Ch. 9 in Introduction to Commutative Algebra. Reading,MA: Addison-Wesley, 1969.

Cohn, H. Introduction to the Construction of Class Fields. New York: Cambridge University Press, p. 32, 1985.

Fröhlich, A. and Taylor, M. Ch. 2 in Algebraic Number Theory. New York: Cambridge University Press, 1991.

Noether, E. "Abstract Development of Ideal Theory in Algebraic Number Fields and Function Fields." Math. Ann. 96, 26-61, 1927.