Let E = {a + bi : a, b ∈ Z, b is even}. 1. A subring S of a ring R is called an ideal of R if sr, rs ∈ S for all s ∈ S and for all r ∈ R. Is S an ideal of Z[i]? Justify answer.

Q: For any ideal A in the ring R, prove that A+A=A

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Q: Let R be a commutative ring and let a ∈ R . Show that I a = { x ∈ R ∣ a x = 0 } is an ideal of R.

A: Given: Let R be a commutative ring and let a ∈ R . To Show that      I a = {x∈R  ax = 0}  is an…

Q: If A is an ideal of a ring R and 1 belongs to A, prove that A = R.

A: Given: A is an ideal of a ring R and 1∈A.To prove: A=R

Q: 1. Let I and J be ideals of a ring R. Prove that IJ is an ideal of R.

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Q: Let I be an ideal of a ring R. Prove that R/I is commutative if and only if ab-ba EI for any a, b €…

A: Let I be an ideal of a ring R. We are to prove that R/I is commutative if and only if ab-ba belongs…

Q: 2. Let B and C be a ideals of a ring R. Prove that the set B - C = {teR such that t = f -g for some…

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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: (b) (-a)b = −(ab) = a(−b) for all a,b € R.

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

A: The objective of the question is to determine whether the union of two ideals I and J of a ring R is…

Q: Let R be a commutative ring with unity and let a, b E R. Show that (a, b), the smallest ideal of R…

A: Here, R be a commutative ring with unity and let a,b∈R. It has to be proved that <a,b>, the…

Q: Show that the union of a chain I, CI,c of ideals of a ring R is an ideal of R. (This exercise is…

A: Given: We are given a chain of ideals in a ring Objective: The objective is to show that the union…

Q: Let R be a ring and let I be an ideal of R. Prove that the factor ring R/I is commutative if and…

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Q: = Let I andJ be two ideals of a commutative ring R. Show that: S= {r€R: ri E J Vi EI } is an ideal…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: Let R be a ring. Let p= ax + b and q = cx² + dx + e be elements of R[x], for a, b, c, d, e E R. True…

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Q: If U, V are ideals of a ring R, let U + V = {u+ v:u E U,v E V}. Prove that U +V is also an ideal.

A: We have to prove the conditions of ideal

Q: Suppose A and B are ideal of the ring R. if A C

A: I have used definition of ideal of ring.

Q: Given that (I,+,.) is an ideal of the ring (R,+,.) Show that : a- the ring (R/I,+,.) may have…

A: We have to show that by example