Let I be an ideals of a ring R and S C R: (i) Show that InS is an ideal of S. Is it also an ideal of R? (ii) Either prove or give counterexample that every ideal of S is of the type InS for some ideal I of R. (iii) Show that if I n S homomorphism. {OR} then 7(S) = S, where T : R → R/I is the projection %3D

Let I be an ideals of a ring R and S C R: (i) Show that InS is an ideal of S. Is it also an ideal of R? (ii) Either prove or give counterexample that every ideal of S is of the type InS for some ideal I of R. (iii) Show that if I n S homomorphism. {OR} then 7(S) = S, where T : R → R/I is the projection %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[x]/I provides an abstract construction of the ring R[ɛ]. (a) Define a function : R[ɛ] → R[x]/I by (a + b) = a + bx + I Verify that is a ring homomorphism. (b) Consider a polynomial f(x) = a + bx + c2x² + €3x³ + ….. +Cnx² € R[x]. Show that f(x) + I = a + bx + I are equal as equivalence classes. (c) Using part (b), explain why the homomorphism from part (a) is surjective.
2
3. Consider P = P(2, v6) = {a + bV6|a, b e Z, 2|a} and Z[V6] = {a + bv6|a, b e Z}. (a) Prove that P is an ideal of the ring Z[V6]. (b) Determine the quotient ring Z[/6]/P. (c) Construct the addition and multiplication tables for the quotient ring Z[V6]/P.
Prove (ii) and (iii)
I have the solution to this question I just need 2-3 examples of this:
Let : R₁ → R₂ be a homomorphism of rings. Show that the mapping o': R₁[x] → R₂[x] given by o'(am +...+α₁x+αo): o(am) +...+ (a₁)x+ o(ao) is a homomorphism of rings with kero' = kero.
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Let R be a commutative ring with unity and let S be a nonempty subset of R. Show that I(S) := {f(x) = R[x] : f(a) = 0, for all a € S} is an ideal of R[x].
Consider the ring R= usual. x -{] Prove that I = a 0 x,yez with matrix addition and multiplication defined as is an ideal of R. (Hint: show that I is a subgroup and then show it is an ideal.)