phism, where R is a commutative ring with unity. Le he arbitrary elements of R. Then there exists a uniqu phism : Fx₁,...,n] →→→ R such that (1) voe=6
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do it in typed
![Theorem 1 [Universal Property] Let : F→ R be a ring homo-
morphism, where R is a commutative ring with unity. Let b₁,..., bn be
some arbitrary elements of R. Then there exists a unique ring homo-
morphism &: F[x1, ..., xn] → R such that (1) vo€ = ¢ (2) ¥(x₁) = bi
voe:
for all i = 1,..., n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff177d89a-9a56-46e5-9efa-67fa59feb238%2F4f2d0e33-8f31-46f6-8c9a-c7a3f2057ac9%2Fodd6u2h_processed.png&w=3840&q=75)
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- A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.Exercises Prove Theorem 5.3:A subset S of the ring R is a subring of R if and only if these conditions are satisfied: S is nonempty. xS and yS imply that x+y and xy are in S. xS implies xS.Let 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=xzyz
- Describe the kernel of epimorphism in Exercise 22. 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.21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.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]22. Let be a ring with finite number of elements. Show that the characteristic of divides .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.
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .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 ].