11. Let I be an ideal in a ring R with unity e. Prove that if e Є I, then I = R. 12 I et I be an ideal in aring R with unity Prove that if I contains an element

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

Exercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .

17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.

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

Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .

Show that the ideal is a maximal ideal of .

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

Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of

33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)