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.
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- 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 .
- 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?