PlanetMath: cancellation ideal

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cancellation ideal

(Definition)

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

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

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 and a regular element of , is cancellative if and only if $\mathfrak{a}$ is cancellative in the multiplicative semigroup of the non-zero integral ideals of .
If $r\in R$ , then the principal ideal of is cancellative if and only if is a regular element of the total ring of fractions of .
If $\mathfrak{a}_1\!+\!\mathfrak{a}_2\!+\!...\!+\!\mathfrak{a}_m$ is a cancellation ideal and a positive integer, then
$\displaystyle (\mathfrak{a}_1\!+\!\mathfrak{a}_2\!+\!...\!+\!\mathfrak{a}_m)^n = \mathfrak{a}_1^n\!+\!\mathfrak{a}_2^n\!+\!...\!+\!\mathfrak{a}_m^n.$
In particular, if the ideal $(a_1,\,a_2,\,...,\,a_m)$ of is cancellative, then
$\displaystyle (a_1,\,a_2,\,...,\,a_m)^n = (a_1^n,\,a_2^n,\,...,\,a_m^n).$

Bibliography

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).

"cancellation ideal" is owned by pahio.

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Other names:

cancellative ideal

Also defines:

cancellative

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Cross-references: ideal, integer, positive, total ring of fractions, principal ideal, integral ideal, iff, product, finite, fractional ideals, semigroup, multiplicative, regular elements, commutative ring
There are 3 references to this entry.

This is version 7 of cancellation ideal, born on 2005-07-18, modified 2005-07-23.
Object id is 7236, canonical name is CancellationIdeal.
Accessed 2358 times total.

Classification:

AMS MSC:

13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)

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