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- Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]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+y4[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [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 ].18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .19. Find a specific example of two elements and in a ring such that and .
- Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]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.