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.
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.
