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Unformatted text preview: PMA346, RINGS AND MODULES, PROBLEMS (1) Let X be any nonempty set. Let R be the set of all subsets of X . For any A,B ∈ R (i.e. A,B ⊆ X ), define A + B = AΔB := ( A ∪ B )  ( A ∩ B ) (the “symmetric difference” of A and B ) and define A · B = A ∩ B . Prove that ( R, + , · ) is a commutative ring. (In particular, identify the additive and multiplicative identities.) Is it a field? (2) Without looking, write down the definitions of a group and a ring. Then look at Definitions 1.1 and 1.2 to see how you did. (3) Let R = M 2 ( R ) , the set of 2 × 2 matrices over R , with the usual addition and multiplication of matrices. You may assume it is a ring. Is it commutative? Let S be the subset fl a b b a ¶ : a,b ∈ R . Show that S is a subring of R . Is it a field? Construct a bijection between S and some wellknown ring, and prove that it is an isomorphism. (4) Let X be any nonempty set, S any ring, and let T = S X , the set of all functions f : X → S . On T , define addition and multiplication in the obvious way, i.e. ( f + g )( x ) := f ( x )+ g ( x ) and ( f · g )( x ) := f ( x ) · g ( x ) (for all x ∈ X ). Prove that this makes T a ring. Show that, with an inspired choice for S , you can get this way something isomorphic to the ring R in Problem 1. (5) Let F 2 = Z /2 Z , the field of 2 elements, and let R = F 2 [ X ] / a , where a = ( X 2 + X + 1 ) . Let α := [ X ] = X = X + a . (a) What quadratic polynomial is satisfied by α ? In terms of α , list all the elements of R . (b) Do tables for addition and multiplication in R . (c) Is R a field? Is R isomorphic to Z /m Z for some integer m ? (d) Let S be any field containing F 2 , with the same number of elements as R . Let β be any element of S not in F 2 . List the elements of S as F 2linear combinations of 1 and β . Which one must β 2 be? Hence produce an isomorphism between R and S . (6) See scan on the PMA346 webpage. (7) State and prove the First Isomorphism Theorem without looking. (8) Let R be a commutative ring. Let T = { ( a ,a 1 ,a 2 ,... ) : all a i ∈ R , and a i = for all but finitely many i } . Any element of T is then of the form a = ( a ,a 1 ,...,a n ,0,0,... ) for some n which in general depends on a . For a = ( a ,a 1 ,... ) and b = ( b ,b 1 ,... ) , define a + b := ( a + b ,a 1 + b 1 ,... ) and a · b := ( a b ,a b 1 + a 1 b ,a b 2 + a 1 b 1 + a 2 b ,a b 3 + a 1 b 2 + a 2 b 1 + a 3 b ,... ) . (a) It is clear that ( T, +) is an abelian group. You may assume this. Prove that the rest of the axioms for a commutative ring are satisfied by ( T, + , · ) . Date : 5th December, 2009. 1 2 PMA346, RINGS AND MODULES, PROBLEMS (b) Can you see a subring of T that is isomorphic to R ? (You needn’t prove it is a subring–just say what it is.) (c) Let X := ( 0,1,0,0,0,... ) ∈ T . What are X 2 and X 3 ?...
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 Winter '11
 PhanThuongCang
 Integral domain, FP, Ring theory

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