In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors...

of a ring Simplicial commutative ring Special types of rings: Boolean ring Dedekind ring Differential ring Exponential ring Finite ring Lie ring Local...

Richard Dedekind Dedekind cut Dedekind domain Dedekind eta function Dedekind-infinite set Dedekind number Dedekind psi function Dedekind sum Dedekind zeta...

In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A...

In mathematics, a ring is said to be a Dedekind-finite ring if ab = 1 implies ba = 1 for any two ring elements a and b. In other words, all one-sided...

not a field. R is a valuation ring with a value group isomorphic to the integers under addition. R is a local Dedekind domain and not a field. R is a...

usage of prime elements in ring theory. A cornerstone of algebraic number theory is, however, the fact that in any Dedekind ring (which includes Z [ − 5...

Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings...

commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of...

Most rings of algebraic integers are not principal ideal domains. This is one of the main motivations behind Dedekind's definition of Dedekind domains...

fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and...

over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal...

number Dedekind's problem Dedekind–Peano axioms Dedekind psi function Dedekind ring Dedekind sum Dedekind valuation Dedekind zeta function Dedekind–Hasse...

domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. The ring of integers...

y} have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition. A lattice ( L , ≤ ) {\displaystyle (L,\leq )} is called...

projective, flat, and torsion-free are equivalent. See local ring, perfect ring and Dedekind ring. Free object Projective object free presentation free resolution...

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which...

of a Dedekind ring is a Dedekind ring. Every overrring of a direct sum of rings whose non-unit elements are all zero-divisors is a Noetherian ring.: 53 ...

characteristic there are Dedekind domains (and even discrete valuation rings) that are not G-rings. Every localization of a G-ring is a G-ring Every finitely generated...

Jacobson ring. Any principal ideal domain or Dedekind domain with Jacobson radical zero is a Jacobson ring. In principal ideal domains and Dedekind domains...

Dedekind rings (the main class of commutative rings occurring in algebraic number theory), integral extensions, and valuation rings. Polynomial rings...

In mathematics, an arithmetic surface over a Dedekind domain R with fraction field K {\displaystyle K} is a geometric object having one conventional dimension...

commutative Artinian ring. If I is a nonzero ideal of a Dedekind domain A, then A / I {\displaystyle A/I} is a principal Artinian ring. For each n ≥ 1 {\displaystyle...

these rings O K {\displaystyle {\mathcal {O}}_{K}} is they are Dedekind domains. Hence the theory of fractional ideals can be described for the rings of...

a G-ring. It is a J-2 ring as all Noetherian local rings of dimension at most 1 are J-2 rings. It is also universally catenary as it is a Dedekind domain...

the polynomial ring R [ x , y ] {\displaystyle \mathbb {R} [x,y]} , although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy...

where P = (π). The previous example can be generalized to Dedekind domains. Let R be a Dedekind domain, K its field of fractions, and let P be a non-zero...

always verified for rings of quadratic integers, as seen above for the case of Z[√−5]. However, as for every Dedekind domain, a ring of quadratic integers...

In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane...

A Dedekind domain (e.g., rings of integers) is a Noetherian domain in which every ideal is generated by at most two elements. The coordinate ring of...