A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K.C . O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective mod...

It is shown that for any commutative unital ring R the category HopfR of R–Hopf algebras is locally presentable and a coreflective subcategory of the category BialgR of R–bialgebras, admitting cofree Hopf algebras over arbitrary R–algebras. The proofs are based on an explicit analysis of the construction of colimits of Hopf algebras, which generalizes an observation of Takeuchi. Essentially be ...

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the th...

We provide necessary and sufficient conditions for a Gaussian ring R to be semihereditary, or more generally, of w.dimR ≤ 1. Investigating the weak global dimension of a Gaussian coherent ring R, we show that the only values w.dimR may take are 0, 1 and ∞; but that fP.dimR is always at most one. In particular, we conclude that a Gaussian coherent ring R is either Von Neumann regular, or semiher...

In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is universal localization. As an application, prove the Telescope Conjecture holds for unbounded derived categories of rings commutative).

It is known that every dimension group with order-unit of size at most א1 is isomorphic to K0(R) for some locally matricial ring R (in particular, R is von Neumann regular); similarly, every conical refinement monoid with order-unit of size at most א1 is the image of a V-measure in Dobbertin’s sense, the corresponding problems for larger cardinalities being open. We settle these problems here, ...

Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matric...