Ring theory is the branch of abstract algebra that studies rings, algebraic structures with an addition and a multiplication that behave much as they do on the integers, without requiring every nonzero element to have a multiplicative inverse. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhich structural properties of a ring, above all its ideals, determine how its elements factor, and how far the patterns familiar from the integers extend to rings that are not commutative. 1 Key DebateHow far the ideal theory Emmy Noether formalized in the 1920s could be pushed toward a full structural classification of rings. The commutative case yields deep results built on that foundation, but noncommutative rings resist as clean a classification, so how far the analogy with the integers can be carried remains an open methodological question rather than a settled one. 1 Classification
Pure or Applied Connections
Associated With
Includes
Source Wolfram MathWorld: Dixmier Conjecture
Source Jacobson's conjecture (Wikipedia)
Source Wikipedia: Jakob Levitzki
Source Köthe conjecture (Wikipedia)
Source MacTutor biography of Øystein Ore, St Andrews
Source Richard Dedekind, Biography (MacTutor History of Mathematics)
Source Wikipedia: Shimshon Amitsur
Source Skolem-Noether theorem, Wikipedia
Sources
1. Ring Theory (Wikipedia)
WikipediaLead section, opening paragraph
ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties
lead paragraph
In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.
History section
In 1920, Emmy Noether, in collaboration with W. Schmeidler, published a paper about the theory of ideals in which they defined left and right ideals in a ring.
View the Source Jacobson's conjecture (Wikipedia)
Includes: Jacobson's Conjecture, Lead sentenceQuote, Includes: Jacobson's Conjecture, Lead sentence
ebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a N
Köthe conjecture (Wikipedia)
Skolem-Noether theorem, Wikipedia
Wikipedia: Shimshon Amitsur
Includes: Shimshon Amitsur, LeadQuote, Includes: Shimshon Amitsur, Lead
best known for his wide-ranging contributions to ring theory
Wikipedia: Jakob Levitzki
Includes: Jakob Levitzki, LeadQuote, Includes: Jakob Levitzki, Lead
for their work on the laws of noncommutative rings
Wolfram MathWorld: Dixmier Conjecture
Wolfram MathWorldIncludes: Dixmier Conjecture, Subject classificationsQuote, Includes: Dixmier Conjecture, Subject classifications
Algebra Ring Theory
MacTutor biography of Øystein Ore, St Andrews
Includes: Øystein Ore, BiographyQuote, Includes: Øystein Ore, Biography
who worked in ring theory, Galois theory and graph theory
Richard Dedekind, Biography (MacTutor History of Mathematics)
Includes: Richard Dedekind, SummaryQuote, Includes: Richard Dedekind, Summary
Richard Dedekind 's major contribution was a redefinition of irrational numbers in terms of Dedekind cuts. He introduced the notion of an ideal in Ring Theory.
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