Mathematics
MA416: Rings (5 ECTS)
An introduction to ring theory, covering topics like PIDs, Polynomial rings, and Euclidean rings.
Taught in Semester(s) 1. Examined in Semester(s) 1.
Workload: 100 hours (24 Lecture hours, 4 Tutorial hours, 72 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Determine whether a given algebraic structure is a ring or not.
- Determine the group of units and the set of zerodivisors in a ring.
- Explain the concepts of homomorphisms, ideals, kernels and quotient rings and relate them to each other.
- Calculate the field of fractions of an integral domain.
- Determine whether a given polynomial is irreducible or not.
- Prove Gauss lemma and Eisenstein's criterion.
- Find the maximal and prime ideals of a given commutative ring.
- Decide whether a given domain is a Euclidean ring.
Indicative Content
An introduction to Ring theory. The material covered includes:
- Basic definitions: rings, units, zerodivisors, fields and integral domains. The group of units of a ring. Fundamental examples of rings: integers, rationals, reals, polynomials, integers modulo $n$.
- Ring homomorphisms: definitions, examples, kernels, images and isomorphisms.
- Ideals and quotient rings: definition of left, right and two-sided ideals, construction of the quotient ring, the first isomorphism theorem.
- Fields of fractions: construction.
- Polynomial rings: irreducibility, primitivity, unique factorisation, Gauss' lemma, and Eisenstein's irreducibility criterion.
- Euclidean rings: definitions, basic properties and case studies of e.g. the Gaussian integers, and Laurent polynomial rings.
Module Resources
"A first course in abstract algebra", John B Fraleigh, Pearson.
"Topics in algebra", I.N. Herstein.
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