4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the quotient ring R/I is a field.
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- 17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.
- 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.If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.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 .
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)