16.2.12. Let I be an ideal in a commutative ring R. Prove that I[x] is an ideal in R[x]. Prove that R[x]/I[x] = (R/I)[x].

Q: Show that f:Z2 -> Z2 defined by f(n) = n2-n is a ring homomarphism.

A: The element in Z2 are {0,1} Now, f(0) = 02 - 0 = 0           f(1) = 12 - 1 = 0 Hence, f maps Z2 to…

Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

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Q: 9. Let R be the ring Q[x]/(x³ - x² − x + 1). (a) List all the ideals of R and draw a diagram showing…

A: It is given that R=Qxx3-x2-x+1. (a) The objective is to list all ideals of R and draw a diagram that…

Q: 2. Prove that I := {f(x) = R[x]: f(1) = 0} is an ideal of R[x] and find a well-known ring R₂ s that…

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Q: To submit Let R be a ring. Define a function e : R[x] × R → R by the rule e(a„x" + ·..+a¡x+a0,r) =…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: a) The ring R[x, y]/(x + 1) is a field. b) The ring Z[x]/(7) is a principal ideal domain

A: a) The ring R[x, y]/(x + 1) is not a field. To determine if it is a field, we need to check if every…

Q: 18.6.1. Let R be a non-trivial commutative ring with identity. Show that the following are…

A: Step 1:(a) => (b):  If ( R ) is local, then it has a unique maximal ideal, denoted by ( M ). The…

Q: (x), J = (x + 1) en K := (x²020 + x) in the polynomial ring R[x]. Rewrite the ideal I · (I + (J^ K))…

A: Aim: To define the principal ideals and in the polynomial ring . And to define as principal…

Q: Find all values of p such that Z„[x]/(x² + 1) is a field.

A: Given problem is :

Q: Let R′ be a commutative ring of characteristic 2. Show that                    I ={r∈R′|r^2 =0} is…

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Q: 1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx J…

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Q: Let R be a unital ring. Let r e R be a nilpotent element. Show that 1R- r e Inv (R). (6) Let 4 E R…

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Q: 1. Prove that if S is a subring of a ring R then the following are true: (a) Os = OR (b) if 1R € S,…

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Q: (a) i. Explain why the ring Z4 is not a field. ii. Explain why the ring Z₂[x]/(x² + 1) is not a…

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Q: Let R and S be commutative rings. Suppose ϕ : R → S is a ring isomorphism. A student says, “By…

A: Let R and S be commutative rings.

Q: Corollary - Let F be a field. Then F[x] is a unique factorization domain.

A: Given: Theorem 18.3: Every principal ideal domain is a unique factorization domain. Corollary: Let F…

Q: Define the following principal ideals of the polynomial ring R[x]: I := (x³ + 1), J := (x + 1) and K…

A: Suppose  that   be  a  polynomial ring and  I = and .Then, IJ=Here, given that   be the  polynomial…

Q: 4. (a) Let S be a commutative ring with 1 and let s E S. Assume that sn = 0 for some integer n ≥ 2.…

A: The objective of this question is to prove that if s is an element of a commutative ring S such that…