A NOTE ON THE CANCELLATION PROPERTIES OF SEMISTAR OPERATIONS

If D is an integral domain with quotient field K, then let F(D) be the set of non-zero D-submodules of K, F(D) be the set of non-zero fractional ideals of D and f(D) be the set of non-zero finitely generated D-submodules of K. A semistar operation * on D is called arithmetisch brauchbar (or a.b.) if, for every H is an element of f(D) and every H-1, H-2 is an element of F(D), (H H-1)* = (H H-2)* implies H-1(*) = H-2(*), and * is called endlich arithmetisch brauchbar (or e.a.b.) if the same holds for every F, F-1, F-2 is an element of f(D). In this note, we introduce the notion of strongly arithmetisch brauchbar (or s.a.b.) and consider relationships among semistar operations suggested by other related cancellation properties.