# 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

 $\mathfrak{ab=ac}\,\Rightarrow\,\mathfrak{b=c}\quad\forall\,\,\mathfrak{b,\,c}% \in\mathfrak{S}.$
• Each invertible ideal is cancellative.

• A finite product $\mathfrak{a}_{1}\mathfrak{a}_{2}...\mathfrak{a}_{m}$ of fractional ideals is cancellative iff every $\mathfrak{a}_{i}$ is such.

• 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 $R$.

• 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 $R$.

• 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 CancellationIdeal 2015-05-06 14:49:08 2015-05-06 14:49:08 pahio (2872) pahio (2872) 11 pahio (2872) Definition msc 13B30 cancellative ideal CancellativeSemigroup IdealDecompositionInDedekindDomain cancellative