(e) Prove that the quotient ring R[x]/(x² - 1) is isomorphic to the product ring RX R. I
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![(e) Prove that the quotient ring R[x]/(x² - 1) is isomorphic to the product ring
RX R.
I](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe0f2e6-a3fe-44ed-a932-d5c5d1248f5a%2F06e31031-30a1-4b1b-8d07-2946d06b6ebb%2Fg4645e_processed.jpeg&w=3840&q=75)
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- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .15. Let and be elements of a ring. Prove that the equation has a unique solution.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?
- 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.)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.Assume that the set R={[x0y0]|x,y} is a ring with respect to matrix addition and multiplication. Verify that the mapping :R defined by ([x0y0])=x is an epimorphism from R to Z. Describe ker and exhibit an isomorphism from R/ker to
- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.
