Hom worke: Consider the ring (Z [√3], +,.), Let A={ a+b√3:a, be Z₂} Is A subring of Z[√3]?
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![Hom worke:
Consider the ring (Z [√3], +,.), Let A={ a+b√3:a, beZ₂}
Is A subring of Z[√3]?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18827c0e-3152-460f-a184-35895ef87b9e%2F8ea1f067-efc8-4a1b-8bc9-49e4c8fd4cad%2Fbvl98fa_processed.jpeg&w=3840&q=75)
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- [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]22. Let be a ring with finite number of elements. Show that the characteristic of divides .Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][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 ].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.18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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+y421. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
- 33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)Let 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.A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.