ABSTRACT
Let M be a module over an associative ring R and σ [M] the category of M-subgenerated modules. Generalizing the notion of a projective generator in σ[M], a module P ∈ σ[M] is called tilting in σ[M] if (i) P is projective in the category of P-generated modules, (ii) every P-generated module is P-presented, and (iii) σ[P] = σ[M]. We call P self-tilting if it is tilting in σ[P]. Examples of (not self-small) tilting modules are ℚ/ℤ in the category of torsion ℤ-modules, ℚ⊕ℚ/ℤ in the category ℤ-Mod, certain divisible modules over integral domains, and also cohereditary coalgebras C over a QF-ring in the category of comodules over C. Self-small tilting modules P in σ[M] are finitely presented in σ[M]. For M = P, they are just the *-modules introduced by C. Menini and A. Orsatti, and for M = R, they are the usual tilting modules considered in representation theory.
Notice that our techniques and most of our results also apply to locally finitely generated Grothendieck categories.
