Ring of Dedekind

In general Algebra, a ring of Dedekind is a ring noethérien completely closed of which any ideal first not no one is maximum, and who is not a body.

That is to say $A$ a commutative ring. The following formulations are equivalent:

$A$ is a ring of Dedekind;
$A$ is a ring of noethérien, integrates, and for any maximum ideal $p$ , the localized of $A$ in $p$ is a ring of discrete valuation;
All the nonnull ideals of $A$ are invertible (this assumption implies in particular that the ring $A$ is just and noethérien).

Here some examples of rings of Dedekind.

principal rings (and thus in particular the ring $\ mathbb {Z}$ of the whole relative and the ring $k$ of the polynomials to coefficients in a commutative body $k$ ).

the ring of the entireties of a algebraic Bodies of numbers.

This last example, which comes from the Algebraic theory of the numbers, is one of most important. It is, from a historical point of view, that which justified the study of the rings of Dedekind. A concrete example is the subring of the body $\ mathbb {C}$ of the complex numbers made up of the elements of the form $a \ sqrt {2} +b$ , with $a$ and $b$ whole relative.

The study of the rings of Dedekind started when Dedekind introduced the concept of ideal into a ring with the hope to compensate for the failure of the single decomposition in factors first in the algebraic whole rings of . While the rings of Dedekind are not factorial in general, they have all the following property: any ideal not no one can be broken up in a single way like a product of ideal first. This explains why Dedekind thought that the ideals were “idealized numbers”.

If we think of the ideals as integers, then the fractional ideal play the part of the fractions. If $A$ is a ring of Dedekind of body of $K$ fractions, a fractional ideal of $A$ is a sous- $A$ -module of the finished type of $K$ , i.e. a sous- $A$ -module $I$ for which there exists an element not no one $r$ of $K$ such as $rI$ is an ideal of $A$ : it is said that $r$ is a common denominator to the elements of $I$ .

The fractional ideals can be added and multiplied like the ordinary ideals, and those differing from zero can be reversed: the reverse of $I$ is the fractional ideal made up of the elements $x$ of $K$ for which $xI$ is included in $A$ . It is then true that $II^ {- 1} =K$ . Any fractional ideal can be written in a single way like a product of ideals first of $A$ and their opposite.

The group of the classes of ideals is the quotient of the group of the nonnull fractional ideals by the sub-group of the principal fractional ideals. Its structure contains important properties of the ring of Dedekind.

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