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- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .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 .)
- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.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.)
- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.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?