Let be a commutative ring containing regular elements and be the multiplicative semigroup of the non-zero fractional ideals of . A fractional ideal of is called a cancellation ideal or simply cancellative, if it is a cancellative element of , i.e. if
Each invertible ideal is cancellative.
A finite product of fractional ideals is cancellative iff every is such.
The fractional ideal , where is an integral ideal of and a regular element of , is cancellative if and only if is cancellative in the multiplicative semigroup of the non-zero integral ideals of .
If , then the principal ideal of is cancellative if and only if is a regular element of the total ring of fractions of .
If is a cancellation ideal and a positive integer, then
In particular, if the ideal of is cancellative, then
- 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).