characterisation

In mathematics, characterisation usually means a property or a condition to define a certain notion.  A notion may, under some presumptions, have different 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  $(a,b)(c,d)=(ac,bd,(a+b)(c+d))$  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.

Title	characterisation
Canonical name	Characterisation
Date of creation	2013-03-22 14:22:28
Last modified on	2013-03-22 14:22:28
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	18
Author	pahio (2872)
Entry type	Definition
Classification	msc 00A05
Synonym	characterization
Synonym	defining property
Related topic	AlternativeDefinitionOfGroup
Related topic	EquivalentFormulationsForContinuity
Related topic	MultiplicationRuleGivesInverseIdeal