define characteristic of a ring. hence find the characteristic of the ring z3 × z4

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]

11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .

Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4

19. Find a specific example of two elements and in a ring such that and .

37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.

A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.

21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.

Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.

[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]