It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) R is right self-injective if R R is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are investigated.

Abstract. In this paper we develop a general theory of modules which are invariant under automorphisms of their covers and envelopes. When applied to specific cases like injective envelopes, pure-injective envelopes, cotorsion envelopes, projective covers, or flat covers, these results extend and provide a much more succinct and clear proofs for various results existing in the literature. Our results are based on several key observations on the additive unit structure of von Neumann regular rings.

No abstract

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had shown that if R is a finite-dimensional algebra over a field F with more than two elements then an indecomposable automorphism-invariant right R-module must be quasiinjective. In this paper we show that this result fails to hold if F is a field with two elements. Dickson and Fuller had further shown that if R is a finitedimensional algebra over a field F with more than two elements, then R is of right invariant module type if and only if every indecomposable right R-module is automorphism-invariant. We extend the result of Dickson and Fuller to any right artinian ring. A ring R is said to be of right automorphism-invariant type (in short, RAI-type) if every finitely generated indecomposable right Rmodule is automorphism-invariant. In this paper we completely characterize an indecomposable right artinian ring of RAI-type.2000 Mathematics Subject Classification. 16U60, 16D50.

This unique and comprehensive volume provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. The finiteness conditions and homological properties are closely interrelated in the sense that either hypothesis induces the other in some form. This is the first book to bring all of this important material on the subject together.