16: Associative rings and algebras
Here are a few notes on (noncommutative) associative ring theory. ( Commutative rings are treated separately, as are non-associative rings). There is a long FAQ on sets with products (rings), a particular emphasis of which is the study of division rings over the reals. Associative division algebras are of particular importance.
This includes the study of matrix rings, division rings such as the quaternions, and rings of importance in group theory. Various tools are studied to enable consideration of general rings.
For detailed expository information you are welcome to to peruse an on-line textbook made freely available by a colleague. Material standard to any beginning course in associative algebras: the Wedderburn structure theorem for semisimple algebras, discussions of the various chain conditions on modules, the Jacobson radical, Artinian algebras, the Krull - Schmidt theorem, the structure of projective modules over Artinian rings, Wedderburn's principal theorem, the Brauer group of a field, and varieties of algebras and polynomial identities.
For the commutative case, See 13-XX
This image slightly hand-edited for clarity. In addition,
the data used for drawing the map are limited to papers since 1991: prior to
that year, there was only one subdivision, 16A, and if the old data are
included, the connections between the detailed subdivisions shown below
and the other branches of math are distorted in the map.
Until 1990 there was only one section, 16A.
Browse all (old) classifications for this area at the AMS.
A good modern graduate-level text is Pierce, Richard S.: "Associative algebras", Graduate Texts in Mathematics, 88. Springer-Verlag, New York-Berlin, 1982. 436 pp. ISBN 0-387-90693-2
Somewhat more comprehensive is Rowen, Louis H.: "Ring theory", Academic Press, Inc., Boston, MA, 1991. 623 pp. ISBN 0-12-599840-6
A thorough review of the modern literature is contained in "Reviews in Ring Theory, 1980-1984", American Mathematical Society, 1986 ISBN 0-8218-0097-3, together with the corresponding assemblage of reviews 1940-1979.
Online notes for a graduate course in ring theory.
A survey article: Bergman, George M. "Everybody knows what a Hopf algebra is", Group actions on rings (Brunswick, Maine, 1984), 25--48; Contemp. Math., 43, Amer. Math. Soc., Providence, R.I., 1985. MR87e:16024