Consider the subset S = {(0,0), (1,1), (2,2), (3,3)} of Z4 × Z4.a. Prove that S is a subring of 74 x 74-b. Is S a commutative ring? Justify your answer.c. Is S a ring with identity? Justify your answer.d. Is S an integral domain? Prove or disprove.e. Is S a field? Prove or disprove.

Answered: Consider the subset S = {(0,0), (1,1),…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 35E: Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a...

See similar textbooks

Similar questions

37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition and multiplication as defined in 18, consider the following questions. Is S a ring? If not, give a reason. Is S a commutative ring with unity? If a unity exists, compare the unity in S with the unity in 18. Is S a subring of 18? If not, give a reason. Does S have zero divisors? Which elements of S have multiplicative inverses?
19. Find a specific example of two elements and in a ring such that and .
Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
True or False Label each of the following statements as either true or false. Every field is a division ring.
Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
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 ].
28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
32. Consider the set . a. Construct addition and multiplication tables for, using the operations as defined in . b. Observe that is a commutative ring with unity, and compare this unity with the unity in . c. Is a subring of ? If not, give a reason. d. Does have zero divisors? e. Which elements of have multiplicative inverses?

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

Consider the subset S = {(0,0), (1,1), (2,2), (3,3)} of Z4 × Z4. a. Prove that S is a subring of 74 x 74- b. Is S a commutative ring? Justify your answer. c. Is S a ring with identity? Justify your answer. d. Is S an integral domain? Prove or disprove. e. Is S a field? Prove or disprove.