## On spectral torsion theories.

##### Abstract

The purpose of this thesis is to investigate how "spect ralness" properties of a torsion theory T on R - Mod are reflected by properties of the ring R and its ring of quotients R,.. The development of "spectral" torsion theory owes much to Zelmanowitz [50] and Gomez-Pardo [23] . Gomez-Pardo proved that there exists a bijective correspondence between the set of spectral torsion theories on R - Modand rings of quotients of R that are Von Neumann regular and left self-injective. Chapter 1 is concerning with the notation used in the thesis and a summary of main results which are needed for understanding the sequel. Chapter 2 is concerned with the construction of a maximal ring of quotients of an arbitrary ring R by using the
notion of denseness and relative injective hull. In Chapter 3, we survey the three equivalent ways of formulating Torsion Theory: by means of preradical functors on the category R- Mod, pairs of torsion / torsion-free classes and topologizing filters on rings. We shall show that Golan's approach to Torsion Theory via equivalence classes of injectives; and Dickson's one (as presented by Stenstrom) are equivalent. With a torsion theory T defined on R-Mod we associate R,. a ring of quotients of R.
The full subcategory (R, T) - Mod of R- Mod whose objects are the T-torsion-free r-injective left R-modules is a Grothendieck category called the quotient category of R - Mod with respect to T. A left R,.-module that is r-torsion-free T-injective as a left R-module is injective if and only if it is injective as a left R-module (Proposition 3.6.4). Because of its use in the sequel , particular attention is paid to the lattice isomorphism that exists between the lattice of .r-pure submodules of a left Rmodule M and the lattice of subobjects of the quotient module M; in the category (R , T) - Mod. Chapter 4 introduces the definition of a spectral torsion theory: a Vll torsion theory r on R - Mod is said to be spectral if the Grothendieck category (R, r) - Mod is spectral. Using the notion of relative essential submodule, one can construct a spectral torsion theory from an arbitrary torsion theory on R - Mod.
We shall show how an investigation of a general spectral torsion theory on R - Mod reduces to the Goldie torsion theory on R/tT (R) - Mod. Moreover, we shall exhibit necessary and sufficient conditions for R; to be a regular left self-injective ring (Theorem 4.2.10). In Chapter 5, after constructing the torsion functor Soce(-) which is associated with the pseudocomplement r.l of r in R - tors, we show how semiartinian rings can be characterized by means of spectral torsion theories: if a spectral torsion theory r on R - Mod is generated by the class of r-torsion simple left R-modules or, equivalently, cogenerated by the class of r-torsion-free simple left R-modules, then R is a left semiartinian ring (Proposition 5.3.2). Chapter 6 gives Zelmanowitz' important result [50]: R; is a semisimple artinian ring if and only if the torsion theory r is spectral and the associated left Gabriel topology has a basis of finitely generated left ideals. We also exhibit results due to M.J.
Arroyo and J. Rios ([4] and [5]) which illustrate how spectral torsion theories can be used to describe when R; is (1) prime regular and left self-injective, (2) a left full linear ring, and (3) a direct product of left full linear rings. We also study the relationship between the flatness of the ring of quotients R; and the r- coherence of the ring R when r is a spectral torsion theory. It is proved that if r is a spectral torsion theory on R - Mod then the following conditions are equivalent: (1) R is left r-coherent; (2) (Rr)R is flat; (3) every right Rr-module is flat as a right R-module (Proposition 6.3.9). This result is an extension of Cateforis' results.