A gg not gh-cancellative semistar operation which is an extension of a star operation

Abstract

Let R be an integral domain with quotient field K , let h (resp., g, f) be the non-zero R -submodules of K (resp., the non-zero fractional ideals of R , the finitely generated non-zero fractional ideals of R ), and let { x, y } be a subset of the set { f, g, h } of symbols. For a semistar operation (cid:63) on R , if ( EE 1 ) (cid:63) = ( EE 2 ) (cid:63) implies E 1 (cid:63) = E 2 (cid:63) for every E ∈ x and every E 1 , E 2 ∈ y, then (cid:63) is called xy-cancellative. Let (cid:63) be a gg-cancellative semistar operation on R which is an extension of a star operation on R . In this paper, we show that (cid:63) need not be gh-cancellative.