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

抄録

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 ★ on R, if (EE₁)^★ = (EE₂)^★ implies E₁^★ = E₂^★ for every E ∈ x and every E₁, E₂ ∈ y, then ★ is called xy-cancellative. Let ★ be a gg-cancellative semistar operation on R which is an extension of a star operation on R. In this paper, we show that ★ need not be gh-cancellative.