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Let D be an integral domain, S be a (not necessarily saturated) multiplicative subset of D, w be the so-called w-operation on D, and M be a unitary D-module. As generalizations of strong Mori domains (respectively, UFDs) and strong Mori modules, we define D to be an S-strong Mori domain (respectively, S-factorial domain) if for each nonzero ideal I of D, there exist an s ∈ S and a w-finite type (respectively, principal) ideal J of D such that s I ⊆ J ⊆ I w ; and M to be an S-strong Mori module if M is a w-module and for each nonzero submodule N of M, there exist an s ∈ S and a w-finite type submodule F of N such that s N ⊆ F ⊆ N w . This paper presents some properties of S-strong Mori domains, S-factorial domains and S-strong Mori modules.

In this article, we characterize t-Prufer modules in the class of faithful multi- plication modules. As a corollary, we also characterize Krull modules. Several properties of a t-invertible submodule of a faithful multiplication module are given.

In this article, we generalize the concepts of several classes of domains (which are related to Krull domains) to torsion-free modules, and show that for a faithful multiplication module M over an integral domain R, M is a Krull module if and only if R is a Krull domain. Then we characterize Krull, Dedekind, and factorial modules.

A. Di Nola & S.Sessa [8] showed that two compact spaces X and Y arehomeomorphic iff the MV -algebras C(X,I) and C(Y,I) of continuous functions defined onX and Y respectively are isomorphic. And they proved that A is a semisimple MV -algebraiff A is a subalgebra of C(X) for some compact Hausdorff space X. In this paper, firstlyby use of functorial argument, we show these characterization theorems. Furthermorewe obtain some other functorial results between topological spaces and MV -algebras.Secondly as a classical problem, we find a necessary and sufficient condition on a givenresiduated l-monoid that it is segmenently embedded into an l-group with order unit. 1. IntroductionAn MV-algebra is a universal algebra (A,+,·,∗,0,1) of (2,2,1,0,0) type suchthat (A,+,0) is an abelian monoid and moreover, x+1 = 1,x ∗∗ = x,0 ∗ = 1,x+x ∗ =1,x·y = (x ∗ +y ∗ ) and x+x ∗ y = y+y ∗ x for all x,y ∈ A. By setting x∨y = x+x ∗ yand x∧y = x(x ∗ +y) we have (A,∨,∧,0,1) as a bounded distributive lattice.The system of MV-algebras is a kind of better system in the sense that closedunder subalgebras, quotients and products and the free MV-algebra with a denu-merable set of generators can be described by MV-algebras of continuous I = [0,1]-valued functions on the Hilbert cube [11]. Furthermore, the variety of MV-algebrasis a Malcev variety and has the congruence regularity [10].In the first part of this paper, we establish a dual-adjunction (η,e) : S ‘ C fromthe subcategory X of Tychonoff spaces of Top into the subcategory A of semi-simple MV-algebras of M

In this paper, Kaplansky-type theorems are given to characterize GCD-monoids and valuation monoids. Also, (unique) r-factorable monoids are defined and it is shown that S is a Krull monoid if and only if S is a unique t-factorable (resp., w-factorable) monoid if and only if S is a t-factorable (resp., w-factorable) t-Prüfer monoid.

In this paper, we introduce the concept of fuzzy star-operations on an integral domain and show that the set of all fuzzy star-operations on the integral domain forms a complete lattice. We also characterize Prufer domains, psuedo-Dedekind domains, (generalized-) greatest common divisor domains, and other integral domains in terms of the invertibility of certain fractionary fuzzy ideals.

Let S be a grading monoid with quotient group q(S) , let F(S) be the set of fractional ideals of S . For A ∈ F(S) , define A w = {x ∈ q(S) \mid J+x \subseteq A for some f.g. ideal J of S with J -1 =S} and A_ \overline w ={x ∈ q(S)\mid J+x \subseteq A for some ideal J of S with J -1 =S} . Then w and \overline w are star-operations on F(S) such that w ≤ \overline w . Using these star-operations, we give characterizations of Krull semigroups and pre-Krull semigroups. Also we show that for every maximal * -ideal P of S , if S P is a valuation semigroup, then * -cancellation ideals are * -locally principal ideals, where * is a star-operation on S of finite character. Finally, we show that S is a pre-Krull semigroup (H-semigroup) if and only if the polynomial semigroup S[x] is a pre-Krull semigroup (H-semigroup).