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- [Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here][Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]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]
- A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.11. 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 .44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.
- 22. Define a new operation of addition in by and a new multiplication in by. a. Is a commutative ring with respect to these operations? b. Find the unity, if one exists.Exercises 2. Decide whether each of the following sets is a ring with respect to the usual operations of addition and multiplication. If it is not a ring, state at least one condition in Definition 5.1a that fails to hold. The set of all integers that are multiples of . The set of all real numbers of the form with and . The set of all real numbers of the form , where and are rational numbers. The set of all real numbers of the form , where and are rational numbers. The set of all positive real numbers. The set of all complex numbers of the form , where (This set is known as the Gaussian integers.) The set of all real numbers of the form with and . The set of all real numbers of the form with and .[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]