Criteria for flatness and injectivity

Journal of Algebra, 2014

Bijective correspondences are established between endofinite injective left modules, endofinite flat right modules, finite collections of minimal noetherian prime ideals, normalized rank functions on left ideals and characters. Endofinite flat modules are identified as flat covers of modules associated to a minimal noetherian prime ideal, while endofinite flat injectives are characterized by localizations with a semiprimary QF-3 quotient ring.

Communications in Algebra, 2006

We prove various extensions of the Local Flatness Criterion over a Noetherian local ring R with residue field k. For instance, if Ω is a complete R-module of finite projective dimension, then Ω is flat if and only if Tor R n (Ω, k) = 0 for all n = 1,. .. , depth(R). In low dimensions, we have the following criteria. If R is onedimensional and reduced, then Ω is flat if and only if Tor R 1 (Ω, k) = 0. If R is twodimensional, then in order for Ω to be flat, it suffices that it is separated, that its projective dimension is finite and that Tor R 1 (Ω, k) = 0. Many of these criteria have global counterparts and in particular, it is shown that the aadic completion of a flat module of finite projective dimension over an arbitrary Noetherian ring is again flat.

Proceedings of the American Mathematical Society, 1975

The question of when certain cyclic flat modules of a ring are injective (and vice versa) is studied. The consequences of the conditions 'flat' and 'injective' on the simple modules of a ring are discussed.

Contemporary Mathematics, 2006

Transactions of the American Mathematical Society, 1974

Transactions of the American Mathematical Society, 1970

In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules-unmixed-is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.

Bulletin of the Australian Mathematical Society

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

1970

Proceedings of the American Mathematical Society, 2003

Let R be a commutative Noetherian ring with nonzero identity and let the injective envelope of R be flat. We characterize these kinds of rings and obtain some results about modules with nonzero injective cover over these rings.