Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially...

a module $m$ is said to be coretractable if there exists a nonzero homomorphism of every nonzero factor of $m$ into $m$. we prove that all right (left) modules over a ring are coretractable if and only if the ring is morita equivalent to a finite product of local right and left perfect rings.

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = C1, (ii) every N-gradable weak V module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including...

We show that each R-module is n-flat (resp., weakly n-flat) if and only if R is an (n,n− 1)-ring (resp., a weakly (n,n− 1)-ring). We also give a new characterization of n-von Neumann regular rings and a characterization of weak n-von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weak n-von Neu...

As a generalization of countably C-approximating posets, the concept of countably QC-approximating posets is introduced. With the countably QC-approximating property, some characterizations of generalized completely distributive lattices and generalized countably approximating posets are given. The main results are as follows: (1) a complete lattice is generalized completely distributive if and...