cancellation ideal

Let $R$ be a commutative ring containing regular elements and $\mathfrak{S}$ be the multiplicative semigroup of the non-zero fractional ideals of $R$ . A fractional ideal $\mathfrak{a}$ of $R$ is called a cancellation ideal or simply cancellative, if it is a cancellative element of $\mathfrak{S}$ , i.e. if

\mathfrak{ab=ac}\,\Rightarrow\,\mathfrak{b=c}\quad\forall\,\,\mathfrak{b,\,c}% \in\mathfrak{S}.

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Each invertible ideal is cancellative.
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A finite product $\mathfrak{a}_{1}\mathfrak{a}_{2}...\mathfrak{a}_{m}$ of fractional ideals is cancellative iff every $\mathfrak{a}_{i}$ is such.
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The fractional ideal $\mathfrak{a}/r:=\{ar^{-1}\!:\,\,\,a\in\mathfrak{a}\}$ , where $\mathfrak{a}$ is an integral ideal of $R$ and $r$ a regular element of $R$ , is cancellative if and only if $\mathfrak{a}$ is cancellative in the multiplicative semigroup of the non-zero integral ideals of $R$ .
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If $r\in R$ , then the principal ideal $(r)$ of $R$ is cancellative if and only if $r$ is a regular element of the total ring of fractions of $R$ .
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If $\mathfrak{a}_{1}\!+\!\mathfrak{a}_{2}\!+\!...\!+\!\mathfrak{a}_{m}$ is a cancellation ideal and $n$ a positive integer, then

$(\mathfrak{a}_{1}\!+\!\mathfrak{a}_{2}\!+\!...\!+\!\mathfrak{a}_{m})^{n}=% \mathfrak{a}_{1}^{n}\!+\!\mathfrak{a}_{2}^{n}\!+\!...\!+\!\mathfrak{a}_{m}^{n}.$

In particular, if the ideal $(a_{1},\,a_{2},\,...,\,a_{m})$ of $R$ is cancellative, then

$(a_{1},\,a_{2},\,...,\,a_{m})^{n}=(a_{1}^{n},\,a_{2}^{n},\,...,\,a_{m}^{n}).$

References

1 R. Gilmer: Multiplicative ideal theory. Queens University Press. Kingston, Ontario (1968).
2 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).

Title	cancellation ideal
Canonical name	CancellationIdeal
Date of creation	2015-05-06 14:49:08
Last modified on	2015-05-06 14:49:08
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Definition
Classification	msc 13B30
Synonym	cancellative ideal
Related topic	CancellativeSemigroup
Related topic	IdealDecompositionInDedekindDomain
Defines	cancellative