Consider the subset D of M (IR) consisting of the diagonal matrices [9]. a. Prove that D is a subring of M(IR). b. Is D a commutative ring? Prove or disprove. c. Is D a ring with identity? Justify your answer. d. Is D an integral domain? Prove or disprove. e. Is D a field? Prove or disprove.

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.

Let S be the set of all 2X2 matrices of the form [x0x0], where x is a real number.Assume that S is a ring with respect to matrix addition and multiplication. Answer the following questions, and give a reason for any negative answers. Is S a commutative ring? Does S have a unity? If so, identify the unity. Is S an integral domain? Is S a field? [Type here][Type here]

15. Let and be elements of a ring. Prove that the equation has a unique solution.

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 .)

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?

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 ].

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?

Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.