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- Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.
- 36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
- Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of22. Let be a ring with finite number of elements. Show that the characteristic of divides .32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .