Let \(R\) be an associative ring with identity and \(\tau\) a hereditary torsion theory on the category of left \(R\)-modules. The purpose of the present book is to offer a presentation of injectivity relative to \(\tau\), with emphasis on the concepts of minimal \(\tau\)-injective modules and \(\tau\)-completely decomposable modules. So the book begins with a general chapter on torsion theories. Chapter 1 mainly contains standard material on torsion theories.
Chapter 2 introduces injectivity relative to \(\tau\) and the notion of relative injective hull of a module.
Let \(M\) be a left \(R\)-module and \(N\) a submodule of \(M\). Then \(N\) is called \(\tau\)-dense (respectively \(\tau\)-closed) in \(M\) if \(M/N\) is \(\tau\)-torsion (respectively \(\tau\)-torsionfree). \(M\) is called \(\tau\)-injective if any homomorphism from a \(\tau\)-dense left ideal of \(R\) to \(M\) can be extended to a homomorphism from \(R\) to \(M\). Furthermore the \(\tau\)-closure of \(N\) in \(M\) is defined as the unique minimal \(\tau\)-closed submodule of \(M\) containing \(N\). A \(\tau\)-injective hull of \(M\) is then defined as the \(\tau\)-closure of \(M\) in \(E(M)\), an injective hull of \(M\).
It is then shown that every left \(R\)-module has a \(\tau\)-injective hull, unique up to isomorphism, and the relationship between \(\tau\)-injectivity and the usual injectivity is analyzed.
Chapters 3 and 4 contain the main results of the book.
A non-zero left \(R\)-module which is the \(\tau\)-injective hull of each of its nonzero submodules is called minimal \(\tau\)-injective. This is the torsion-theoretic analogue of indecomposable injective modules and plays an important part in direct sum decompositions of \(\tau\)-injective modules. Chapter 3 deals with minimal \(\tau\)-injective modules and shows the structure of the \(\tau\)-injective hull of a module.
A direct sum of minimal \(\tau\)-injective submodules is called \(\tau\)-completely decomposable. Chapter 4 developes a study of \(\tau\)-completely decomposable modules and investigates when \(\tau\)-injective modules are \(\tau\)-completely decomposable.
A left \(R\)-module \(M\) is called \(\tau\)-complemented if every submodule of \(M\) is \(\tau\)-dense in a direct summand of \(M\). It is shown that the class of \(\tau\)-completely decomposable modules is a subclass of the class of \(\tau\)-complemented modules and that some results on \(\tau\)-completely decomposable modules give partial answers to a question generalizing a problem of Matlis.
A left \(R\)-module \(M\) is called \(\tau\)-quasi-injective if whenever \(N\) is a \(\tau\)-dense submodule of \(M\), every homomorphism \(N\to M\) extends to an endomorphism of \(M\). Chapter 5, the final chapter of the book, deals with \(\tau\)-quasi-injective modules. Some of their properties discuss relationships between certain conditions on \(\tau\)-injectivity and \(\tau\)-quasi-injectivity for modules in the context of \(\tau\)-natural classes. Here a \(\tau\)-natural class means a class of left \(R\)-modules closed under isomorphic copies, submodules, direct sums and \(\tau\)-injective hulls.
Throughout the book, the proofs are careful and many examples are offered.