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MATH21112 Rings And Fieldsexample Sheet 7More On Ideals1. Show That If R Is Any Ring Then R Is A Division Ring If The Only Right Ideals Of R Are {0} And R. (A Similar Statement Is True For Left Ideals).2. Let R = Z And Let I = &<3&>. What Is I2 ? What Is In? Let J = &<12&>. What Is IJ?3. Let I,
2024/6/20 10:49:30
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MATH21112 Rings And Fields Example Sheet 6 More On Homomorphisms And Ideals 1. Show That The Map From Z[√2] To Itself Given By Sending A + B√2 Toa -B√2 Is An Automorphism. Show That There Are No Other Automorphisms Of Z[√2] Apart From The Identity Map.2. Let R = Z [√2] And Let Define The Map Pr
2024/6/19 10:56:32
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MATH21112 Rings And Fieldsexample Sheet 5Homomorphisms And Isomorphisms1. Prove That Isomorphism Is An Equivalence Relation On The Set Of All Rings.2. Show That The Ring Of 2 × 2 Matrices Of The Form.With A,B,C, D ∈ R (And I A Square Root Of —1) Is Isomorphic To The Ring Of Quaternions (See Examp
2024/6/18 12:01:03
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MATH21112 Rings And Fieldsexample Sheet 4Fields, Nilpotents And Idempotents1. Show That Q[I] = {A + Bi J A, B ∈ Q} (Where I2 = -1) Is A field.2. Let R Be The Polynomial Ring Z8 [X]. Show That The Polynomial 1 + 2X Is Invertible In R.(Hint: Consider Powers Of 1 + 2X.)3. Let R Be A Commutative Rin
2024/6/18 9:27:39
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MATH21112 Rngs And Fieldsexample Sheet 2 - Rings And Subrings1. Let R Be The Set Of Functions F : R → R With Addition And Multiplication Defined As(F + G)(X) = F(X) + G(X) And (Fg)(X) = F(X)G(X) For All X ∈ R.Show That R Is A Ring By Verifying The Ring Axioms (R1)-(R4).2. Show That Z[√2] = {A +
2024/6/17 13:48:42
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MATH21112 Rings And Fieldsexample Sheet 3 - Domains1. Prove That If R Is A Domain Then Char(R) = 0 Or Char(R) Is A Prime Integer.2. Is Mn (R) A Domain?3. Prove That In A Domain R The Cancellation Rule Applies Ie. For All A,B, C ∈ R, If Ab = Ac And A ≠ 0 Then B = C.Give An Example O
2024/6/17 13:48:42
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MATH21112 Rings And Fieldsexample Sheet 1 - Properties Of Z And Zn1. Find The Greatest Common Divisor Of 2827 And 374 And Write The Gcd As A Linear Combination Of 2827 And 374.2. Let N ∈ Z With N ≥ 2. Prove That 三 Mod N Has The Following Properties:(I) 8A ∈ Z, A 三 A Mod N.(Ii) 8A, B ∈ Z, If A 三 B
2024/6/17 10:07:05
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