WHEN THE SEMISTAR OPERATION ˜ ⋆ IS THE IDENTITY GIAMPAOLO PICOZZA AND FRANCESCA TARTARONE Abstract. We study properties of integral domains in which it is given a semistar operation ⋆ such that ˜ ⋆ is the identity. In particular, we put attention to the case ⋆ = v, where v is the divisorial closure. 1. Introduction Throughout D is an integral domain with quotient field K. To avoid trivial cases we assume that D is not a field. We denote by F(D) the set of nonzero D-modules contained in K, by f (D) the set of nonzero finitely gen- erated D-modules contained in K and by F(D) the set of nonzero fractional ideals of D. A semistar operation on D is a map ⋆ : F (D) → F (D),E → E ⋆ , such that, for all x ∈ K, x = 0, and for all E,F ∈ F (D), the following properties hold: (⋆ 1 ) (xE) ⋆ = xE ⋆ ; (⋆ 2 ) E ⊆ F implies E ⋆ ⊆ F ⋆ ; (⋆ 3 ) E ⊆ E ⋆ and E ⋆⋆ := (E ⋆ ) ⋆ = E ⋆ . (cf. [33] and [16]). In the following we denote by d D (or simply by d) the identity semistar operation on F(D). We say that a semistar operation is trivial if E ⋆ = K for each E ∈ F(D) and this happens if and only if D ⋆ = K (in fact, for each E ∈ F(D), E = ED, whence E ⋆ =(ED) ⋆ ⊇ ED ⋆ = K and so E ⋆ = K). If ⋆ is a semistar operation on D, we can consider a map ⋆ f : F (D) → F (D) defined as follows: for each E ∈ F (D), E ⋆ f :=  {F ⋆ | F ⊆ E, F ∈ f (D)}. It is easy to see that ⋆ f is a semistar operation on D, called the semistar operation of finite type associated to ⋆. Note that, for each F ∈ f (D), F ⋆ = F ⋆ f . A semistar operation ⋆ is called a semistar operation of finite type if ⋆ = ⋆ f . Moreover (⋆ f ) f = ⋆ f (that is, ⋆ f is of finite type). A quasi–⋆–ideal of D is a nonzero ideal I such that I = I ⋆ ∩ D.A quasi– ⋆–prime is a quasi–⋆–ideal that is also a prime ideal. A quasi–⋆–maximal ideal is an ideal that is a maximal element in the set of quasi–⋆–prime ideals. If ⋆ is a semistar operation of finite type, each quasi-⋆-ideal is contained in 2000 Mathematics Subject Classification. Primary: 13A15, 13F05, 13G05. Key words and phrases. Semistar operation, w-operation, t-linked overrings. 1