Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.
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- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .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.Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal 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.)27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- 15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .