A Note on the Cancellation Properties of Semistar Operations

If D is an integral domain with quotient field K, then let F̄(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 ∈ F̄(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. Let D be an integral domain with quotient field K. Let F̄(D) be the set of non-zero D-submodules of K and let F(D) be the set of non-zero fractional ideals of D (i.e., E ∈ F(D) if E ∈ F̄(D) and there is a non-zero element d ∈ D with dE ⊂ D). We also let f(D) be the set of non-zero finitely generated D-submodules of K. A star operation on D is a mapping ? : F̄(D) −→ F̄(D) G 7−→ G? such that, for every x ∈ K−{0} and every G,G1, G2 ∈ F̄(D), the following properties hold: (1) (x)? = (x), (2) (xG)? = xG?, (3) G1 ⊂ G2 implies G1 ⊂ G2, (4) G ⊂ G?, and (5) (G?)? = G?. A star operation ? on D is called arithmetisch brauchbar (or a.b.) if, for every F ∈ f(D) and every G1, G2 ∈ F(D), (FG1) = (FG2) implies G1 = G2, and ? is called endlich arithmetisch brauchbar (or e.a.b.) if the same holds for every F, F1, F2 ∈ f(D). A semistar operation on D is a mapping ? : F̄(D) −→ F̄(D) H 7−→ H?, such that, for every x ∈ K−{0} and every H1, H2 ∈ F̄(D), properties (2) through (5) above hold. Similar to the situation above, a semistar operation ? on D is called a.b. if, for every 2000 Mathematics Subject Classification. Primary 13A15.