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Let R be a domain of Krull dimension one. We study when the class F of modules over R that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If R is local, we show that F is closed under direct summands if and only if any indecomposable, finitely generated, torsion-free module has local endomorphism ring. If, in addition, R is noetherian, this is equivalent to saying that the normalization of R is a local ring. If R is an h -local domain of Krull dimension 1 and F_R is closed under direct summands, then the property is inherited by the localizations of R at maximal ideals. Moreover, any localization of R at a maximal ideal, except maybe one, satisfies that any finitely generated ideal is 2 -generated. The converse is true when the domain R is, in addition, integrally closed, or noetherian semilocal, or noetherian with module-finite normalization. Finally, over a commutative domain of finite character and with no restriction on the Krull dimension, we show that the isomorphism classes of countably generated modules in F are determined by their genus.

We use pullbacks of rings to realize the submonoids MM of (N0∪{∞})k(\\mathbb {N} _0\\cup \\{\\infty \\})^k, which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right RR-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left RR-modules is determined by the monoid D(M)D(M) defined by reversing the inequalities determining the monoid MM. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters. We also provide a rich variety of examples of semilocal rings having nonfinitely generated projective modules that are finitely generated modulo the Jacobson radical.

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a question posed by A.~Amini, B.~Amini, M.~Ershad and H.~Sharif (2007).

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.

This notes aims to clarify the proof given by Raynaud and Gruson that the Mittag-Leffler property descents via pure rings monomorphism of commutative rings. A consequence of that is that projectivity dencents via such ring homomorphisms, a revision of the proof also allows to prove that the property of being pure-projective also descents via pure monomorphisms between commutative rings.

We make a systematic approach to (strict) Mittag-Leffler inverse system and to dual Mittag-Leffler direct systems. This allows us to prove that a right We also study when classes defined via vanishing either of Finally we also show that suitable classes of relative Mittag-Leffler modules give new examples of non deconstructible classes and, over countable rings, they give new examples of non precovering classes.

We prove that every tilting module of projective dimension at most one is of finite type, namely that its associated tilting class is the Ext-orthogonal of a family of finitely presented modules of projective dimension at most one.

We thoroughly investigate the trace ideals of projective modules over the endomorphism ring of a uniserial module. After the work of Dubrovin and Puninski, it is known that this class of rings provides examples of trace ideals of projective right modules that are not trace ideals of projective left modules. In this paper we further investigate when this happens, giving an intrinsic description of such trace ideals and their properties. We also use the theory associated to lifting projective modules modulo a trace ideal to give an alternative approach to Puninski's construction of a direct summand of a serial module that is not serial.

Let RR be a commutative domain. We prove that an RR-module BB is projective if and only if ExtR1(B,T)=0\\mathrm {Ext}_R^1(B,T)=0 for any torsion module TT. This answers in the affirmative a question raised by Kaplansky in 1962.

We study local morphisms in the setting of general noncommutative rings. In particular, we apply local morphisms to study endomorphism rings of modules. We use our constructions to determine classes of modules with semilocal endomorphism rings. For instance, we prove that every finitely presented right module over a semilocal ring has a semilocal endomorphism ring.