# 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 Characterisation 2013-03-22 14:22:28 2013-03-22 14:22:28 pahio (2872) pahio (2872) 18 pahio (2872) Definition msc 00A05 characterization defining property AlternativeDefinitionOfGroup EquivalentFormulationsForContinuity MultiplicationRuleGivesInverseIdeal