R is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV-ring, then R is right uniform or a right V-ring. It is shown that a right WV-ring R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS (complements are summands, a.k.a. ‘extending modules’) or noetherian module. For a finitely generated module M with projective socle over a V-ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X ⊕ T, where X is semisimple and T is noetherian with zero socle. In the case where M = R, we get R = S ⊕ T, where S is a semisimple artinian ring and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.

We call a module Malmost perfect if every M-generated flat module is M-projective. Any perfect module is almost perfect. We characterize almost-perfect modules and investigate some of their properties. It is proved that a ring R is a left almost-perfect ring if and only if every finitely generated left R-module is almost perfect. R is left perfect if and only if every (projective) left R-module is almost perfect.

A module M is called poor whenever it is N-injective, then the module N is semisimple. In this paper the properties of poor modules are investigated and are used to characterize various families of rings.

In this paper we consider for a non-unital ring R, the category of firm R-modules for a non-unital ring R, i.e. the modules M such that the canonical morphism μM : R ⊗RM → M given by r ⊗ m ↦ rm is an isomorphism. This category is a natural generalization of the usual category of unitary modules for a ring with identity and shares many properties with it. The only difference is that monomorphisms are not always kernels. It has been proved recently that this category is not Abelian in general by providing an example of a monomorphism that is not a kernel in a particular case. In this paper we study the lattices of monomorphisms and kernels, proving that the lattice of monomorphisms is a modular lattice and that the category of firm modules is Abelian if and only if the composition of two kernels is a kernel.

We recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.

Algebras A and coalgebras C over a commutative ring R are defined by properties of the (endo)functors A ⊗R – and C ⊗R – on the category of R-modules R. Generalising these notions, monads and comonads were introduced on arbitrary categories, and it turned out that some of their basic relations do not depend on the specific properties of the tensor product. In particular, the adjoint of any comonad is a monad (and vice versa), and hence, for any coalgebra C, HomR(C, –), the right adjoint of C ⊗R –, is a monad on R. The modules for the monad HomR(C, –) were called contramodules by Eilenberg–Moore and the purpose of this talk is to outline the related constructions and explain the relationship between C-comodules and C-contramodules. The results presented grew out from cooperation with G. Böhm, T. Brzeziński and B. Mesablishvili.

If the injective hull E = E(RR) of a ring R is a rational extension of RR, then E has a unique structure as a ring whose multiplication is compatible with R-module multiplication. We give some known examples where such a compatible ring structure exists when E is a not a rational extension of RR, and other examples where such a compatible ring structure on E cannot exist. With insights gleaned from these examples, we study compatible ring structures on E, especially in the case when ER, and hence RR ⊆ ER, has finite length. We show that for RR and ER of finite length, if ER has a ring structure compatible with R-module multiplication, then E is a quasi-Frobenius ring under that ring structure and any two compatible ring structures on E have left regular representations conjugate in Λ = EndR(ER), so the ring structure is unique up to isomorphism. We also show that if ER is of finite length, then ER has a ring structure compatible with its R-module structure and this ring structure is unique as a set of left multiplications if and only if ER is a rational extension of RR.