A NOTE ON THE CANCELLATION PROPERTIES OF SEMISTAR OPERATIONS

Year 2010, Volume: 7 Issue: 7, 78 - 86, 01.06.2010

S. T. Chapman Ryûki Matsuda

Abstract

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 ∈ f(D) and every H1, H2 ∈ F( ¯ D), (HH1)? = (HH2)? implies H?1 = H?2, and ? is called endlich arithmetisch brauchbar (or e.a.b.) if the same holds for every F, F1, F2 ∈ 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.

Keywords

semistar operation