5. Prove if u is a unit in a ring R with identity, prove that u is not a zero divisor. [Hint: Use proof by contradiction.]

If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.

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.)

Prove that if R is a field, then R has no nontrivial ideals.

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

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

Prove that a finite ring R with unity and no zero divisors is a division ring.

Label each of the following statements as either true or false. Every subring of a ring R is an idea of R.

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

50. Let and be nilpotent elements that satisfy the following conditions in a commutative ring: Prove that is nilpotent. for some