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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 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. )Assume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.Assume 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.