Given a ring R and a ER. Let S = {ra|r ER}. Construct an example which shows S need not be an ideal, even if R has an identity.

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

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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: 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: Give an example of ring elements a and b with the properties thatab = 0 but ba ≠ 0.

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

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Q: If the rings R and S are isomorphic, show that R[x] and S[x] areisomorphic.

A: We have given that rings R and S are isomorphic. Let us assume that, ϕ : R→S be ring isomorphism.…

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…

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Q: A ring may have more than one identity element for multiplication? T or F

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Q: Let R be a ring. For n greater than or equal to 0, let In = {a, an element of R | 5na = 0}. Show…

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Q: Let a and b be elements of a ring R. Prove that the equation a+ x= b has a unique solution.

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Q: Let R be a ring and a, b in R. Show that ab is nilpotent implies that ba is nilpotent.

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A: In abstract algebra, a ring is a set equipped with two binary operations, addition and…

Q: Let R be a ring and a, be R. Show that ab is nilpotent implies that ba is nilpotent.

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

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: Let a and b belong to a ring R and let m be an integer. Prove that m (ab) = (ma)b = a(mb).

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

A: I have used definition of ideal of ring.

Q: Let R be a ring, and suppose that 0 = 1 in R. Show that Ris the zero ring. In other words: show that…

A: Consider the ring  with  in .To show that  is the zero ring.That is to show that if  is any element…