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- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].18. Let be the smallest subring of the field of rational numbers that contains . Find a description for a typical element of .
- Exercises If and are two ideals of the ring , prove that is an ideal of .Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?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 .)
- Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].8. Prove that the characteristic of a field is either 0 or a prime.