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- 14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.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?18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
- 14. Let be an ideal in a ring with unity . Prove that if then .The splitting field of f=(x^2-2)(x^2+1) over ℚ is ℚ(sqrt(2),i) because its roots are sqrt(2),-sqrt(2),i,-i. The Galois group of f contains the following automorphisms: sigma σ: sqrt(2) ↦ -sqrt(2) and tau τ: i ↦ -i. Which subfield of the splitting field is fixed by sigma?If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring operations? How would you prove that? If not, what could be a counter-example?
- a. Is the ring 2Z isomorphic to the ring 3Z?b. Is the ring 2Z isomorphic to the ring 4Z?a) The ring R[x, y]/(x + 1) is a field.b) The ring Z[x]/(7) is a principal ideal domain. please if able explain each taken step in detail, I'm quite new to abstract algebra, thank you in advance.Find the characteristic of the ring (Z/20Z) /(5Z/20Z)
- Derive that (x + y)(x' + z)(y + z) = (x + y)(x' + z) by usingboolean algebra.Let R and S be commutative rings. Suppose ϕ : R → S is a ringisomorphism. A student says, “By plugging a polynomial f from R[x] into ϕyou get an isomorphism from R[x] to S[x].” This is not correct. Explain theerror.Let S[x] = {a+bx where a,b in ℝ} (S[X] is ring) and ℝ<+, *, 0, 1>. Prove that there is no isomorphism between S[X] and ℝ rings or give a counter example.