nLab Dedekind ring

A commutative unital ring $R$ is a Dedekind ring if it is

• Noetherian,
• integrally closed ring,
• and every non-zero prime ideal in $R$ is maximal.

If $R$ is a Dedekind ring then every ideal can be uniquely factored into prime ideals and the non-zero fractional ideals form a group under multiplication of ideal?s.

• Serge Lang, Algebraic number theory, GTM 110, Springer 1970, 2000