Ideals, radicals, and structure of additive categories
Simple and semisimple additive categories are studied. We prove, for example, that an artinian additive category is (semi)simple iff it is Morita equivalent to a division ring(oid). Semiprimitive additive categories (that is, those with zero radical) are those which admit anoether full, faithful functor into a category of modules over a division ringoid.
Mathematics subject classifications (1991)18E05 16A20 16A40
Key wordsAdditive category semiprimitive ring artinian simple Jacobson radical density arguments
Unable to display preview. Download preview PDF.
- 1.Karlheinz Baumgartner: Structure of additive categories,Cahiers topologie et géométrie différentielle 16 (1975), 169–213.Google Scholar
- 2.Karlheinz Baumgartner: General Wedderburn theorems and density for small categories (Preprint, Voitsberg, Austria, March 1987).Google Scholar
- 3.H. Cartan and S. Eilenberg:Homological Algebra, Princeton, 1956.Google Scholar
- 4.B. J. Day and R. H. Street: Categories in which all strong generators are dense,J. Pure Appl. Algebra 43 (1986), 235–242.Google Scholar
- 5.I. N. Hernstein:Noncommutative Rings, The Carus Math Monographs 15, Math. Assoc. of America, 1968.Google Scholar
- 6.Nathan Jacobson:Basic Algebra II, W. H. Freeman & Co., San Francisco, 1980.Google Scholar
- 7.G. M. Kelly: On the radical of a category,J. Australian Math. Soc. 4 (1964), 299–307.Google Scholar
- 8.Barry Mitchell: Rings with several objects,Advances in Math. 8 (1972), 1–161.Google Scholar
- 9.Ross Street: Enriched categories and cohomology,Quaestiones Math. 6 (1983), 265–283.Google Scholar