Let k be a commutative ring, H a faithfully flat Hopf algebra with bijective antipode, A a k-flat right H-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants B = A , and we study the semisimplicity of the category of relative Hopf modules.

We study Gorenstein flat objects in the category Rep(Q,R) of representations a left rooted quiver Q with values Mod(R), all R-modules, where R is an arbitrary associative ring. show that representation X if and only for each vertex i canonical homomorphism φiX:⊕a:j→iX(j)→X(i) injective, R-modules X(i) CokerφiX are flat. As application, we obtain model structure on which give explicit descriptio...

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...