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& L

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]

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.&

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&

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

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&

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& W

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