We use the concept of dual Goldie dimension and a characterization of semi-local rings due to Camps and Dicks (1993) to find some classes of modules with semi-local endomorphism ring. We deduce that linearly compact modules have semi-local endomorphism ring, cancel from direct sums and satisfy the n th root uniqueness property. We also deduce that modules over commutative rings satisfying AB5* also cancel from direct sums and satisfy the n th root uniqueness property. Let R be an associative ring with 1 and let Af be a right unital i?-module. A finite set Ax, ... , An of proper submodules of M is said to be coindependent if for each /, 1 < i < n, A¡ + f\j:jii Aj = M, and a family of submodules of M is said to be coindependent if each of its finite subfamilies is coindependent. The module M is said to have finite dual Goldie dimension if every coindependent family of submodules of M is finite. It can be shown that, in this case, there is a maximal coindependent family of submodules of Af. If this set is finite, then its cardinality (denoted by codim(Af)) is uniquely determined and is called the dual Goldie dimension of Af. If this set is infinite we set codim(Af) = oo and say that Af has infinite dual Goldie dimension. A module with dual Goldie dimension 1 is said to be hollow, and a cyclic hollow module is said to be local. We have codim(Afi © Af2) = codim(Afi) + codim(Af2), codim(Af/7V) < codim(Af) for every submodule N of M, codim(M/N) = codim(Af) if N is a small submodule of Af, codim(Af) = 0 if and only if Af = 0; refer to [10] and [20] for details concerning the dual Goldie dimension. A ring R with Jacobson radical J(R) is said to be semi-local if R/J(R) is a semi-simple ring. Semi-local rings are characterized as those rings with finite dual Goldie dimension. Note that for a semi-local ring R, codim(i?Ä) = length of the right R-module R/J and so codim(i?/;) = codim(/?i?) ; this common value is denoted by codim(i?).

If an automorphism a on a ring R (with 1) leaves no non-trivial proper ideals of R invariant then we say that R is σ-simple. We construct examples of σ-simple rings and prove that finitely generated σ-simple algebras over fields are regular. A geometric interpretation of these concepts is also discussed.

Harada calls a ring R right simple-injective if every R-homomor-phism with simple image from a right ideal of R to R is given by left multiplication by an element of R. In this paper we show that every left perfect, left and right simple-injective ring is quasi-Frobenius, extending a well known result of Osofsky on self-injective rings. It is also shown that if R is left perfect and right simple-injective, then R is quasi-Frobenius if and only if the second socle of R is countably generated as a left R-module, extending many recent results on self-injective rings. Examples are given to show that our results are non-trivial extensions of those on self-injective rings. A ring R is called quasi-Frobenius if R is left (and right) artinian and left (and right) self-injective. A well known result of Osofsky [15] asserts that a left perfect, left and right self-injective ring is quasi-Frobenius. It has been conjectured by Faith [9] that a left (or right) perfect, right self-injective ring is quasi-Frobenius. This conjecture remains open even for semiprimary rings. Throughout this paper all rings R considered are associative with unity and all modules are unitary R-modules. We write M R to indicate a right R-module. The socle of a module is denoted by soc(M). We write N ⊆ M (N ⊆ ess M) to mean that N is a submodule (essential) of M. For any subset X of R, l(X) and r(X) denote, respectively, the left and right annihilators of X in R. A ring R is called right Kasch if every simple right R-module is isomorphic to a minimal right ideal of R. The ring R is called right pseudo-Frobenius (a right PF-ring) if R R is an injective cogenerator in mod-R; equivalently if R is semiperfect, right self-injective and has an essential right socle. A ring R is called right principally injective if every R-morphism from a principal right ideal of R into R is given by left multiplication. In [14], a ring R is called a right generalized pseudo-Frobenius ring (a right GPF-ring) if R is semiperfect, right principally injective and has an essential right socle. We write J = J(R) for the Jacobson radical of the ring R. Following Fuller [10], if R is semiperfect with a basic set E of primitive idempotents, and if e, f ∈ E, we say that the pair (eR, Rf) is an i-pair if soc(eR) ∼ = f R/f J and soc(Rf) ∼ = Re/Je.

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