Abelian categories with applications to rings and modules

- N. Popescu,
**Abelian categories with applications to rings and modules**, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp

Contents:

**Note to the reader**

Some terminology, notations and conventions used throughout the book.

**Categories and functors**

The notion of a category; examples. Special objects and morphisms. Functors. Inductive and projective limits. Adjoint functors; equivalence of categories.

**Abelian categories**

Preadditive and additive category. The canonical factorization of a morphism; preabelian categories. Abelian categories. Fibred products and fibred sums. Basic lemmas on abelian categories. The isomorphism theorems. Direct sums of subobjects. Inductive limits; the conditions Ab.

**Additive functors**

Additive functors. Exactness of functors; injective and projective objects. The injective and projective objects in the category Ab. Categories of additive functors; modules. Special objects in abelian categories. Tensor products; a characterization of functor categories. Tensor products of modules. Flat modules. Some remarks on projective modules. Injective envelopes. Semisimple rings.

**Localization**

Categories of fractions; calculus of fractions. The spectral category of an abelian category. The quotient category of an abelian category relative to a dense subcategory. The section functor. Localization in categories with injective envelopes. Localization in Ab 3 categories. Sheaves over a topological space. Torsion theories. Localizing subcategories in categories of modules. Localization in categories of modules. Left exact functors; the embedding theorem. The study of the localization ring of a ring. The complete ring of quotients. Some remarks on Grothendieck categories. Finiteness conditions on localizing systems. Flat epimorphism of rings. Left quasi-orders of a ring. Rings of fractions. Left orders. Left spectrum of a ring. Bilocalizing subcategories.

**The Krull-Remak-Schmidt theorem and decomposition theories**

The classical Krull-Remak-Schmidt theorem. The structure of spectral categories. Local coirreducible categories. Decomposition theory. Semi-noetherian categories. Semi-artinian categories. Noetherian and artinian categories. Locally noetherian categories. Noetherian and artinian rings. Primary decomposition theory. Decomposition theories on locally noetherian categories.

**Duality**

Linearly compact subcategories. Topologically linearly compact rings. The duality theorem for Grothendieck categories. Duality theory for l.n.- and l.f.-categories. Colocalization.

**Bibliography**

**Index**

Created on April 8, 2015 at 17:47:03. See the history of this page for a list of all contributions to it.