Let R be a commutative ring such that 2x =% xe R. Then the mapping f : R → R defined as f (x) = x is a homomorphism. %3D
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- 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+y424. 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 .)14. Let be an ideal in a ring with unity . Prove that if then .
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .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?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 ].
- 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.)[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][Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]