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- 52. (See Exercise 51.) a. Write out the elements of and construct addition and multiplication tables for this ring. (Suggestion: Write for, for in.) b. Is a commutative ring? c. Identify the unity elements, if one exists. d. Find all units, if any exist. e. Find all zero divisors, if any exist. f. Find all idempotent elements, if any exist. g. Find all nilpotent elements, if any exist. Exercise 51. 51. Let and be arbitrary rings. In the Cartesian product of and, define if and only if and , , . Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of and and is denoted by. Prove that is commutative if both and are commutative. Prove has a unity element if both and have unity elements. Given as example of rings and such that does not have a unity element.39. (See Exercise 38.) Show that the set of all idempotent elements of a commutative ring is closed under multiplication. Exercise 38. An element in a ring is called idempotent if . Find two different idempotent elements in .Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]
- 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.An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().
- 14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.40. Let be idempotent in a ring with unity. Prove is also idempotent.
- Given that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.Prove that a finite ring R with unity and no zero divisors is a division 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?