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# characterisation

In mathematics, characterisation usually means a property or a condition to define a certain notion. A notion may, under some presumptions, have different equivalent ways to define it.

For example, let $R$ be a commutative ring with non-zero unity (the presumption). Then the following are equivalent:

(1) All finitely generated regular ideals of $R$ are invertible.

(2) The formula $(a,\,b)(c,\,d)=(ac,\,bd,\,(a+b)(c+d))$ for multiplying ideals of $R$ is valid always when at least one of the elements $a$, $b$, $c$, $d$ of $R$ is not zero-divisor.

(3) Every overring of $R$ is integrally closed.

Each of these conditions is sufficient (and necessary) for characterising and defining the Prüfer ring.

Related:

AlternativeDefinitionOfGroup, EquivalentFormulationsForContinuity, MultiplicationRuleGivesInverseIdeal

Synonym:

characterization, defining property

Type of Math Object:

Definition

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

00A05*no label found*

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