Show that an intersection of ideals of a ring R is again an ideal of R.

Q: is a maximal ideal of Q[z].

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Q: Consider the ring ℤ12. List the elements of the following principal ideals: (i) ⟨4⟩  (ii) ⟨9⟩…

A: Given ring is ℤ12.

Q: Find two examples of a ring in which {0R} is maximal.

A: Answer

Q: Give an example of a commutative ring R with unity and a maximalideal I of R such that I[x] is not a…

A: In this question,  we have to give an example of a commutative ring R with unity and a maximal ideal…

Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

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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: If a ring R has characteristic zero, then R must have an infinite number of elements. true or false

A: We have find that the given statement  "If a ring R has characteristic zero, then R must have an…

Q: (8) Show that for a field F, the set S of all all matrices of the form a b (88) 00 for a, b F is a…

A: An ideal is a special kind of subring of a ring R. If a∈I and r∈R implies that ra∈I then, the…

Q: Give an example of a ring that has exactly two maximal ideals

A: To Do: Give an example of a ring that has exactly two maximal ideals.

Q: Let K, I and J be ideals of ring R such that both I and J are subsets of K with I C J. Then show…

A: Given:- K,I and J be ideals of Ring R Such that I⊂J⊂K (i) Claim: K/I is subring of R/I ∀r.s∈R K⊂R…

Q: Is it true or false that if I is a principal ideal in a commutative ring R with multiplicative…

A: Is it true or false that if  is a principal ideal in a commutative ring  with multiplicative…

Q: Exercise 36. Let A,B, and C be ideals of a commutative ring R. Prove the following: (1) A.(B.C)…

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Q: 4. Prove that a zero divisor in a ring cannot be a unit.

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Q: given an example for a ring R has ideal I such that : VI #I + I c VI

A: Ring:- A ring is a set R with two operations say "+, * " has the following properties:- 1. R is a…

Q: (1) Let a be a fixed element in the commutative ring R, and let A- (xe R: xa = 0). Prove that A is…

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Q: Let R be a ring. For a subset X C R, let (X) denote the ideal generated by X, i.e., the smallest…

A: I have given detailed solution below thanks Explanation:Step 1: Step 2: Step 3: Step 4:

Q: 2. Determine whether or not the set is an ideal in the ring M2×2(R). J-{(8₂9) | PCR} = ER

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: A principal ideal ring is a ring with the property that every ideal hasthe form <a>. Show that…

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Q: C. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

A: NOTE:Hi! Thank you for your question. Since,we only answer 1 question in case of multiple question,…

Q: Let I and J be ideals of a ring R: (i) Show that I+J is the smallest ideal containing both I and J.…

A: The answer is given below

Q: Q) Let (H,+10, -10) be a subring of the ring (Z10, +10. -10 ), where H = {[0], [2], [4], [6], [8]}.…

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Q: Let P be a prime ideal and R a commutative ring with identity. If R/P is an integral domain, explain…

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Q: Prove that 2/3Z Z, as rings,

A: We shall be using First Isomorphism theorem of rings to solve this problem.

Q: is union of two ideal rings R, an idea of R? prove or give counter example

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Q: The ring Zp2, has exactly-----------maximal ideals

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