Let M and N be ideals of a ring R and let H = {m+n | m∈ M, n ∈ N} (a) Show that H is an ideal of R. (b) Show that M ⊆ H and N⊆ H.

Q: Let R be a ring and consider R x Z = { (r,n) | r e R, ne Z} Define (r,n) + (s,m) =(rts, n+ m) (r,…

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Q: Determine whether T={(a, - a) | aEZ} is a subring of ZxZ.

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

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Q: (Q1)Find all idempotent elements and nilpotent elements of the following rings 2- (Z/(3),+..) + (C.…

A: We know that both given ring are field

Q: a Let Z[√2] = {a+b√√2 | a, b € Z} and let H = { b Show that Z[√2] and H are isomorphic as rings. 2b…

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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: Show that A is ideal of A+B? IS an

A: According to questionWe have to show that A is an ideal of A+B

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: Exercise 3: Let R be a commutative ring. (a) Let I and J be ideals in R. Show that In J is also an…

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Q: aw- Let Z[/2] = {a +b/2 |a, b e Z} and let H = { b а 2b : a,b eZ }. a Show that Z[ /2] and H are…

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Q: Let (R, +,.) be any ring. Define R × Z = {(r,n):r E R ,n E Z} and %3D (r, n) + (s, m) = (r + s,m +…

A: In the Structure T=ℝ×ℤ additional multiplication is defined as…

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: Define principal ideal ring.

A: Ideal: A subring A of a ring R is called a (two-sided) ideal of A if for every r∈R and every a∈A…

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: Let I and J be ideals of a ring R. Prove or disprove (by counterexample) that the following are…

A: Given that I and J are two ideals of a ring R Ideal Test: A nonempty subset A of a ring R is an…

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: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

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

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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: explain if the following statement is true or false in detail: "The ideal (x + 1) is maximal in the…

A: "The ideal (x + 1) is maximal in the ring R[x, y]"

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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Q: In any ring R show that the set S={x∈R|r(x)⋂B≠(0) for all nonzero right ideals B or R} is an ideal…

A: To show that the set S = {x ∈ R | r(x) ∩ B ≠ (0) for all nonzero right ideals B of R} is an ideal in…

Q: Find all the principal ideals of the ring (Z6, O, 0 0 ).

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Q: Let I,J be the ideals of a ring P. Show that the sets (a)lJ = {a,b, +.….+a,b, :a, E1,b, E…

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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: Prove that Q[x]/<x2 - 2> is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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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: Let I and J be ideals of a ring R.Prove that I+J= , the ideal of R generated by I ∪ J.

A: I and J are the ideals of a ring R.