3. Suppose that 4 : R → S and : S→T are ring homomorphisms. Recall that the compositionof the functions and is the function o: RT defined by(v op)(a) = (y(a)) for all a € RShow that oy is a ring homomorphism.

Answered: Image /qna-images/answer/88ffea85-58e6-4af1-8874-29ff5cb2aafe.jpg

Answered: Suppose that : R Sand : S→T are ring…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 14E: 14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.

See similar textbooks

Similar questions

Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]
14. Let be an ideal in a ring with unity . Prove that if then .
14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.
Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]
6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
Exercises If and are two ideals of the ring , prove that is an ideal of .
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Exercises 18. Suppose and let be defined by . Prove or disprove that is an automorphism of the additive group .

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

Suppose that : R Sand : S→T are ring homomorphisms. Recall that the composition of the functions and is the function 04: R→T defined by (vy) (a) = (y(a)) for all a € R Show that op is a ring homomorphism.