We study the abelian category W (p)-mod of modules over the triplet W algebra W (p). We construct the projective covers P± s of all the simple objects X± s , 1 ≦ s ≦ p, in the category W (p)-mod. By using the structure of these projective modules, we show that W (p)-mod is a category which is equivalent to the abelian category of the finite-dimensional modules for the restricted quantum group Ū...

In this paper, we study the left orthogonal class of max-flat modules which are homological objects related to s-pure exact sequences and module homomorphisms. Namely, a right $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any $R$-module $B$, strongly ${Ext}^{i}_{R}(A,B)=0$ all $R$-modules $B$ $i\geq 1$. Firstly, give some properties $MF$-projective SMF-projective modules. Then intro...

an r-module m is called epi-retractable if every submodule of mr is a homomorphic image of m. it is shown that if r is a right perfect ring, then every projective slightly compressible module mr is epi-retractable. if r is a noetherian ring, then every epi-retractable right r-module has direct sum of uniform submodules. if endomorphism ring of a module mr is von-neumann regular, then m is semi-...