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Unformatted text preview: Rings and Modules Old Syllabus for O4 T. W. K¨ orner October 5, 2004 Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). Please note that, throughout, ring means commutative ring with one. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This document is written in L A T E X2e and stored in the file labelled ~twk/1B/Rings.tex on emu in (I hope) read permitted form. My email address is twk@dpmms . Contents 1 Rings 2 2 Ideals, quotients and the isomorphism theorem 4 3 Integral domains, fields and fractions 6 4 Unique factorisation, Euclidean and principal ideal domains 11 5 Polynomials over rings 14 6 Unique factorisation for polynomials 18 7 Fields and their simple extensions 22 8 Splitting fields of polynomials 26 9 Finite fields 28 10 Modules 31 1 11 Linear relations in modules 35 12 Matrices and modules 38 13 The module decomposition theorems 42 14 Applications to endomorphisms 47 15 Reading and further reading 51 1 Rings The same ideas and proofs occur in the study of the integers (number theory), polynomials (leading to algebraic geometry), parts of the theory of matrices and in the theory of Abelian groups. They may be unified by using the theory of commutative rings and modules following a programme laid out by Emmy Noether and others. We start by looking at commutative rings with one. Definition 1 We say that ( R, + ,. ) is a commutative ring with a one if (i) ( R, +) is an Abelian group. (ii) a ( bc ) = ( ab ) c for all a,b,c ∈ R . [Associative law of multiplication.] (iii) a ( b + c ) = ab + ac , ( b + c ) a = ba + ca for all a,b,c ∈ R . [Distributive law.] (iv) There exists a 1 ∈ R such that 1 a = a 1 = a for all a ∈ R . [Existence of a multiplicative identity.] (v) ab = ba for all a,b ∈ R . [Commutative law of multiplication.] Rules (iii) and (iv) could be shortened using rule (v). We usually write 0 for the identity of the group ( R, +) and call 0 the zero of R . Rule (iv) is made easier to use by the following simple remark. Lemma 2 (Uniqueness of multiplicative identities) If ( M,. ) is an ob ject with multiplication and 1 , 1 ∈ M are identities in the sense that 1 a = a 1 = a and 1 a = a 1 = a for all a ∈ M , then 1 = 1 . Thus R has a unique multiplicative identity 1. (We shall usually refer to 1 as ‘one’. It is sometimes called ‘the unit element of R ’ but the word ‘unit’ means something different in the context of this course, see Definition 42.) 2 There are important examples of noncommutative rings (that is systems obeying all the rules in Definition 1 except (v) the commutative law of mul tiplication) such as the set of n × n matrices with the usual addition and multiplication [ n ≥ 2]. However, there are many beautiful results which are only true for commutative rings. Rule (iv) (the existence of a one) is lessonly true for commutative rings....
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 Fall '11
 DrewArmstrong
 The Land

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