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Unformatted text preview: Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the secondyear course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester 2005–2006. After a short introductory chapter consisting mainly of reminders about such topics as functions, equivalence relations, matrices, polynomials and permuta tions, the notes fall into two chapters, dealing with rings and groups respec tively. I have chosen this order because everybody is familiar with the ring of integers and can appreciate what we are trying to do when we generalise its prop erties; there is no wellknown group to play the same role. Fairly large parts of the two chapters (subrings/subgroups, homomorphisms, ideals/normal subgroups, Isomorphism Theorems) run parallel to each other, so the results on groups serve as revision for the results on rings. Towards the end, the two topics diverge. In ring theory, we study factorisation in integral domains, and apply it to the con struction of fields; in group theory we prove Cayley’s Theorem and look at some small groups. The set text for the course is my own book Introduction to Algebra , Ox ford University Press. I have refrained from reading the book while teaching the course, preferring to have another go at writing out this material. According to the learning outcomes for the course, a studing passing the course is expected to be able to do the following: • Give the following. Definitions of binary operations, associative, commuta tive, identity element, inverses, cancellation. Proofs of uniqueness of iden tity element, and of inverse. • Explain the following. Group, order of a group, multiplication table for a group, subgroups and subgroup tests, cyclic subgroups, order of an element, Klein four group. • Describe these examples: groups of units of rings, groups of symmetries of equilateral triangle and square. • Define right cosets of a group and state Lagrange’s Theorem. Explain nor mal subgroup, group homomorphism, kernel and image. iii iv • Explain the following: ring, types of rings, subrings and subring tests, ide als, unit, zerodivisor, divisibility in integral domains, ring homomorphism, kernel and image. Note: The pictures and information about mathematicians in these notes are taken from the St Andrews History of Mathematics website: http://wwwgroups.dcs.stand.ac.uk/ ~ history/index.html Peter J. Cameron April 13, 2006 Contents 1 Introduction 1 1.1 Abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Sets, functions, relations . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Equivalence relations and partitions . . . . . . . . . . . . . . . . 6 1.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Rings 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Introduction ....
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 Spring '10
 CIPERIANI,M
 Algebra, Addition, Ring, Rings, Ring theory

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