Invariance of Modules under Automorphisms of their Envelopes and Covers

Abstract

The theory of invariance of modules under automorphisms of their envelopes and covers has opened up a whole new direction in the study of module theory. It offers a new perspective on generalizations of injective, pure-injective and flat-cotorsion modules beyond relaxing conditions on liftings of homomorphisms. This has set off a flurry of work in the area, with hundreds of papers using the theory appearing in the last decade. This book gives the first unified treatment of the topic. The authors are real experts in the area, having played a major part in the breakthrough of this new theory and its subsequent applications. The first chapter introduces the basics of ring and module theory needed for the following sections, making it self-contained and suitable for graduate students. The authors go on to develop and explain their tools, enabling researchers to employ them, extend and simplify known results in the literature and to solve longstanding problems in module theory, many of which are discussed at the end of the book.

Remarks on separativity of regular rings

“…A module M is called pseudo injective if every monomorphism from a submodule N of M can be extended to an endomorphism of M ( [7]). It has been shown in [6] that M is pseudo injcetive if and only if M is invariant under automorphism of E(M) where E(M) is the injective hull of M (See the recent book [14] ). In view of this, pseudo injective modules are called auto invariant modules.…”

MacWilliams extending conditions and quasi-Frobenius rings

“…. , k, E(P i ) must be a direct sum of indecomposable modules Q j with End(Q j )/J(End(Q j )) ∼ = F 2 , by [23,Corollary 4.29]. And each Q j contains a simple module by hypothesis.…”

“…, C n } is a representative set of the isomorphism classes of the simple left R-modules. We know from [23,Theorem 4.23] that if P i is not quasi-injective, then End(P i )/J(End(P i )) ∼ = F 2 . So we may order the set Λ as follows:…”

“…As P i ⊕ P i ′ is automorphism-invariant, P i ′ is P i -injective and P i ′ is P i -injective (see e.g. [23,Corollary 4.6]). Therefore, ϕ and ϕ −1 extend to homomorphisms f : P i → P i ′ and g :…”

“…This class of modules was first studied by Dickson and Fuller in [4] and it has been studied extensively in recent years. See [23], [9], [10], [11] for more details on automorphism-invariant modules. If R is a ring such that R R is a (quasi-)injective module then R is called a left self-injective ring, whereas if R R is an automorphism-invariant module then R is called a left automorphism-invariant ring.…”