Ivo Herzog

Left Cotorsion Rings

It is proved that if R is an associative ring that is cotorsion as a left module over itself, and J is the Jacobson radical of R, then the quotient ring R/J is a left self-injective von Neumann regular ring and idempotents lift modulo J. In particular, if R is indecomposable, then it is a local ring. Let R be an associative ring with identity. A left R-module RM is called cotorsion if Ext 1(F, M) = 0 for every flat left R-module RF . Cotorsion modules were introduced by Harrison [4], and are a generalization of pure-injective modules. Their interest derives from the recent result of Bican, El Bachir and Enochs (see [1] and [8, Theorem 3.4.6]), that every module admits a cotorsion envelope. The ring R is called left cotorsion if it is cotorsion when considered as a left R-module over itself. Left cotorsion rings are closely related to pure-injectivity from another point of view. The category R -Flat of flat left R-modules is a locally finitely presented additive category, and therefore admits a theory of purity, as propounded in [2]. By [5, Lemma 3], R is pure-injective as an object of R -Flat if and only if it is left cotorsion. The point of this paper is to extend to these kinds of rings, results of Zimmermann and Zimmermann-Huisgen [10] about rings that are pure-injective as left modules over themselves. It is proved (see Theorem 6) that if R is a left cotorsion ring and J is the Jacobson radical of R, then the quotient ring R/J is a von Neumann regular ring. Furthermore, we show (see Corollary 9) that R/J is injective as a left module over itself, and that idempotents lift modulo J (see Corollary 4). Every left pure-injective ring R may be realized as the endomorphism ring of an injective object of some Grothendieck category, so these results about left cotorsion rings may be seen as generalizations of classical results concerning the endomorphism ring of an injective module. However, let us remark that our results are a proper extension, since there exist many left cotorsion rings that are not left pure-injective. Take, for example, any left perfect ring that is not left pureinjective (see [9]). We should also point out that, as the endomorphism ring of any injective object of a Grothendieck category is always left pure-injective, standard functor category techniques cannot be used in the study of left cotorsion rings. We develop in this paper new techniques based on the behaviour of direct limits, whose ¯ ¯

PURE-INJECTIVE ENVELOPES

It is proved that if C is a locally finitely presented additive category, then every object X of C admits a pure-injective envelope. Considered as a morphism, the pure-injective envelope is a pure-monomorphism.

Powers of the phantom ideal

The mono-epi (ME) exact structure on the morphisms of an exact category (A;E) is introduced and used to prove ideal versions of Salces Lemma, Christensens (Ghost) Lemma, and Wakamatsus Lemma for an exact category. Salces Lemma establishes a bijective correspondence I↦I⊥ between the class of special precovering ideals of (A;E) and that of its special preenveloping ideals. ME-extensions of morphisms are used to define an extension I⋄J of ideals. Christensens Lemma asserts that the class of special precovering (respectively, special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salces Lemma satisfies (IJ)⊥=J⊥⋄I⊥ and (I⋄J)⊥=J⊥I⊥. Wakamatsus Lemma asserts that if a covering ideal I is closed under ME-extensions, then it is a special precovering ideal. As an application, it is proved that if G is a finite group and Φ is the ideal of phantom morphisms in the category k[G]-Mod, then Φn−1 is the object ideal generated by projective modules, where n is the nilpotency index of the Jacobson radical J. If R is a semiprimary ring, with Jn=0, then Φn is generated by projective modules. For a right coherent ring R over which every cotorsion left R-module has a coresolution of length n by pure injective modules, Φn+1 is generated by flat modules.

Pure Projective Approximations

A notion of good behavior is introduced for a definable subcategory of left R -modules. It is proved that every finitely presented left R -module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod- R , Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add( R -mod) of pure projective left R -modules. An example is given of a preenveloping subcategory ⊆ Add( R -mod) that does not arise from a covariantly finite subcategory of finitely presented left R -modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R -module is pure injective, then the smallest definable subcategory ( R -proj) containing every finitely generated projective module is well-behaved.

WHEN COTORSION MODULES ARE PURE INJECTIVE

We characterize rings over which every cotorsion module is pure injective (Xu rings) in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts.

Model-theoretic aspects of Σ-cotorsion modules

Abstract Let R be an associative ring with identity. It is shown that every Σ -cotorsion left R -module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ -cotorsion. It follows that, for countable R , the class of Σ -cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the modules M for which Ext 1 ( F , M ( ℵ 0 ) ) = 0 , for every countably presented flat module F .

The nonstandard quantum plane

Abstract Let U q be the quantum group associated to s l 2 ( k ) with char ( k ) ≠ 2 and q ∈ k not a root of unity. The article is devoted to the model-theoretic study of the quantum plane k q [ x , y ] , considered as an L ( U q ) -structure, where L ( U q ) is the language of representations of U q . It is proved that the lattice of definable k -subspaces of k q [ x , y ] is complemented. This is deduced from the same result for the U q -module M , which is defined to be the direct sum of all finite dimensional representations of U q . It follows that the ring of definable scalars for the quantum plane is a von Neumann regular epimorphic ring extension of the quantum group U q .

Modules with regular generic types. Part IV

In this paper, we continue the line of investigation undertaken in [HR], with which we assume familiarity. For a comprehensive reference we refer the reader to [P]. The authors would like to thank each other for their light spirit and good humor. Pillay and Prest [PP, Proposition 7.10] have shown that a module M of U -rank 1 which is not totally transcendental may be decomposed as M = M l ⊕ M u , where M l , ⊨ Th( M ) omits the unlimited type and M u imbeds purely into a model of the unlimited part T u , of T = Th( M ). We devote the first section of this paper to a generalization of this result to the case when T has a regular generic and m-dim( M ) = 1. (Note that m-dim( M ) = 0 implies M is totally transcendental, in which case a general decomposition theorem was proved by Garavaglia [G].)