A module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.

As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

We show that a semiperfect right FPF-ring is right self-injective if and only if J(R) — Z(RR), extending a well-known result due to Carl Faith on semiperfect right FPF-rings with nil Jacobson radical.

It is shown that a two-sided continuous ring with ascending chain condition on left annihilators is quasi-Frobenius.

A ring R is said to be an absolute subretract if for any ring S in the variety generated by R and for any ring monomorphism f from R into S, there exists a ring morphism g from S to R such that gf is the identity mapping. This concept, introduced by Gardner and Stewart, is a ring theoretic version of an injective notion in certain varieties investigated by Davey and Kovacs.

Also recall that a special principal ideal ring is a local principal ring with nonzero nilpotent maximal ideal. In this paper (finite) special principal ideal rings that are absolute subretracts are studied.

Let V be a variety of rings and let A ∊ V. The ring A is injective in V if every triangle

with C ∊ V, m a monomorphism and f a homomorphism has a commutative completion as indicated. A ring which is injective in some variety (equivalently, injective in the variety it generates) is called injective. When only triangles with f surjective are considered we obtain the notion of weak injectivity. Directly indecomposable injective and weakly injective rings are classified.

We prove two theorems on continuous modules: Decomposition Theorem. A continuous module M has a decomposition, M = M1 ⊕ M2, such that M1 is essential over a direct sum of indecomposable summands Ai of M, and M2 has no uniform submodules; and these data are uniquely determined by M up to isomorphism. Direct Sum Theorem. A finite direct sum of indecomposable modules Ai is continuous if and only if each Ai is continuous and Aj-injective for all j ≠ i.