aa_hw_8_Math 113 - Introduction to Abstract Algebra...

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Unformatted text preview: Introduction to Abstract Algebra Homework 8 November 1, 2010 Problem 1 (a) R → R × R Í defined by r Ô→ ( r, 0) is a ring homomorphism, since for any a,b ∈ R , φ ( a + b ) = ( a + b, 0) = ( a, 0) + ( b, 0) = φ ( a ) + φ ( b ), and φ ( ab ) = ( ab, 0) = ( a, 0)( b, 0) = φ ( a ) φ ( b ). (b) R → R × R defined by r Ô→ ( r,r ) is a ring homomorphism, since for any a,b ∈ R , φ ( a + b ) = ( a + b,a + b ) = ( a,a ) + ( b,b ) = φ ( a ) + φ ( b ), and φ ( ab ) = ( ab,ab ) = ( a,a )( b,b ) = φ ( a ) φ ( b ). (c) R × R Í → R defined by ( r 1 ,r 2 ) Ô→ r 1 is a ring homomorphism, since for any ( a,b ) ∈ R and ( c,d ) ∈ R Í , φ (( a,b ) + ( c,d )) = φ (( a + c,b + d )) = a + c = φ (( a,b )) + φ (( c,d )), and φ (( a,b )( c,d )) = φ (( ac,bd )) = ac = φ (( a,b )) φ (( c,d )). (d) R × R → R defined by ( r 1 ,r 2 ) Ô→ r 1 r 2 is not a ring homomorphism, since for any ( a,b ) ∈ R and ( c,d ) ∈ R Í , φ (( a,b ) + ( c,d )) = φ (( a + c,b + d )) = ab + ad + cb + cd Ó = ab + cd = φ (( a,b )) + φ (( c,d )). (e) R × R → R defined by ( r 1 ,r 2 ) Ô→ r 1 + r 2 is not a ring homomorphism, since for any ( a,b ) ∈ R and ( c,d ) ∈ R Í , φ (( a,b )( c,d )) = φ (( ac,bd )) = ac + bd Ó = ac + ad + bc + bd = ( a + b )( c + d ) φ (( a,b )) φ (( c,d )). Problem 2 For some subset I of Z [ x ], for I to be an ideal, we require that rI ⊆ I for all r ∈ Z [ x ], and we require that I is a subring, i.e. ( I, +) < ( Z [ x ] , +) and I is closed under multiplication....
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This note was uploaded on 10/15/2011 for the course CHEM 1 taught by Professor Gelfand during the Summer '11 term at Solano Community College.

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aa_hw_8_Math 113 - Introduction to Abstract Algebra...

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