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).