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Chapter 6 Solutions
Elements Of Modern Algebra
- 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 ].arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forward
- 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 .)arrow_forward22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forwardA Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.arrow_forward
- 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.)arrow_forward17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.arrow_forward11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .arrow_forward
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]arrow_forwardLet R be a ring, and let x,y, and z be arbitrary elements of R. Complete the proof of Theorem 5.11 by proving the following statements. a. x(y)=(xy) b. (x)(y)=xy c. x(yz)=xyxz d. (xy)z=xzyz Theorem 5.11 Additive Inverses and Products For arbitrary x,y, and z in a ring R, the following equalities hold: (x)y=(xy) b. x(y)=(xy) (x)(y)=xy d. x(yz)=xyxz (xy)z=xzyzarrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,