Let F be a commutative ring with identity and let f(x) E F[x] be a polynomial of degree 6. Which of the following statements is true: f(x) has exactly 6 roots counting multiplicity. O f(x) has exactly 6 distinct roots f(x) has at most 6 roots.
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- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.11. 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 .
- 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.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.)An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.
- 12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .True or False Label each of the following statements as either true or false. 3. The characteristic of a ring is zero if is the only integer such that for all in.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]
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.