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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 16: Associative rings and algebras

Introduction

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.

History

Applications and related fields

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.

Subfields

• 16B: General and miscellaneous
• 16D: Modules, bimodules and ideals
• 16E: Homological methods and results, see also 18GXX
• 16G: Representation theory of rings and algebras
• 16H05: Orders and arithmetic, separable algebras, Azumaya algebras, See also 11R52, 11R54, 11S45, 13A20
• 16K: Division rings and semisimple Artin rings, see also 12E15, 15A30
• 16L: Local rings and generalizations
• 16N: Radicals and radical properties of rings
• 16P: Chain conditions, growth conditions, and other forms of finiteness
• 16R: Rings with polynomial identity
• 16S: Rings and algebras arising under various constructions
• 16U: Conditions on elements (including elements of matrix rings, etc.)
• 16W: Rings and algebras with additional structure
• 16Y: Generalizations, For nonassociative rings, see 17-XX
• 16Z05: Computational aspects of associative rings (See also 68W30). [new in 2000]

Until 1990 there was only one section, 16A.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

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

Software and tables

Other web sites with this focus

• This list of ring theorists provides links to other sites of interest.
• Here are the AMS and Goettingen resource pages for area 16.

Selected topics at this site

• Are there (associative, distributive) rings in which the addition is not commutative?
• A practical(?) application of the embedding of matrix rings M_n(C) into M_2n(R).
• What are the endomorphisms of the matrix ring M_n(R)?
• What is a free module and what are some modules that aren't free?
• Rings with isomorphic modules of different ranks (dimension shifting in K-theory)
• How can one define determinants in M_n(A) if A is non-commutative?
• What are the multiplicative scalar functions on matrices? (determinants...)
• Goldie's theorem (noncommutative fields of fractions)
• Can one do number theory within the quaternions?
• Doing number theory in the ring of quaternions.
• What are the integral quaternions?
• Citation and remarks about solving linear equations in the ring of quaternions
• Solving polynomial equations in the ring of quaternions; passing to extension rings.
• Does the Fundamental Theorem of Algebra hold for extension rings of the reals? (Other than the complex numbers, no).
• Pointer and citation to solving quadratic equations in the ring of quaternions.
• What are the (other) roots of p(X)=0 in the ring M_n(F) where p is the characteristic polynomial of a matrix A?
• Family of division algebras over a field (plus Milnor conjecture, K-theory)
• What is the Brauer Group and how does it classify associative algebras over a field?
• Subalgebras of Hopf algebras.
• Subalgebras of matrix algebras.