Show that every ring can be embedded in a ring with identity.
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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.22. Let be a ring with finite number of elements. Show that the characteristic of divides .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 .
- A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.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.)21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
- 15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .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 ].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]