We then begin to study the question of factorization in rings, and find a class of rings, known as Unique Factorization Domains, where any element can be written uniquely as a product of prime elements generalising the case of the integers. Finally, we study modules, which roughly means we study linear algebra over certain rings rather than fields.
This turns out to have powerful applications to ordinary linear algebra and to abelian groups. Students should become familiar with rings and fields, and understand the structure theory of modules over a Euclidean domain along with its implications.
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The material underpins many later courses in algebra and number theory, and thus should give students a good background for studying these more advanced topics. Zero-divisors, integral domains.
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The characteristic of a ring. Discussion of fields of fractions and their characterization proofs non-examinable . Homomorphisms of rings. Quotient rings, ideals and the first isomorphism theorem and consequences, e.
Chinese remainder theorem. Prime ideals and maximal ideals, relation to fields and integral domains. Examples of ideals.
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Euclidean Domains. Principal Ideal Domains. EDs are PIDs.
A Course in Ring Theory
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Abstract: First published in , this book contains the core material for an undergraduate first course in ring theory. Volume: Publication Month and Year: Copyright Year: Page Count: Cover Type: Hardcover. Print ISBN Online ISBN Online ISSN:. Primary MSC: Secondary MSC: Applied Math?
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