PMA346problems

# PMA346problems - PMA346 RINGS AND MODULES PROBLEMS(1 Let X...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PMA346, RINGS AND MODULES, PROBLEMS (1) Let X be any non-empty 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 well-known ring, and prove that it is an isomorphism. (4) Let X be any non-empty 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 2-linear 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 ?...
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

PMA346problems - PMA346 RINGS AND MODULES PROBLEMS(1 Let X...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online