Branches of Mathematics
Ring Theory
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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.
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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 Cross-Tradition Connections
Sources
1. Ring Theory (Wikipedia)
WikipediaLead section, opening paragraphQuote, Lead section, opening paragraph
ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties
1. Ring Theory (Wikipedia)
Wikipedialead paragraphQuote, 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.
1. Ring Theory (Wikipedia)
WikipediaHistory sectionQuote, 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.
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