If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

Q: IF J is nil left ideal in an Artinian ring R, then J is

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Q: Let R be a commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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Q: Let R be a ring with unity. Show that (a) = { £ xay : x, y e R }. finite

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Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

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Q: If the ring R has a left identity as Well right identity then these two as are equal.

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Q: The ring Zs(i) has no proper ideals True False

A: Given statement is false. Justification is given in step two.

Q: 12. Prove that the rings Z,[x]/(x² + x + 2) and Z,[x]/(x² + 2x + 2) are isomorphic.

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Q: If Ø: R S is a ring homomorphism. The Ø preserves:

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Q: In the ring Z;[x]': 1- The inverse of x is:

A: Answer :

Q: Give an example of a non-commutative ring R without unity such that (xy)^2 = x^2.y^2 for all x,y in…

A: We consider the example of a non-commutative ring R without unity such that (xy)2 = x2y2 for all x…

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

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Q: If R is a ring and f(x) and g(x) are of degrees 3 and 4 then f(x)g(x) must have degree 7.

A: Given that R is a ring. deg{f(x)}=3 deg{g(x)}=4

Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: The ring Z-[i] has no proper ideals True False

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Q: 3. Explain why the polynomial rings R[r] and C[r] are not isomorphic.

A: This is a problem of Abstract Algebra.

Q: If in a ring R every x E R satisfies x2 = x, Prove that R must be commutative.

A: Answer and explanation is given below...

Q: Show that if R is a ring with unity and N is an ideal of R such that N R, then R/N is a ring with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

Q: Let R be a ring with unity. Show that (a) = { E xay : x, y e R }. finite

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Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

A: Ring (definition) Let R be a non empty set together with two binary operations called addition(+)…

Q: Let R be a ring with 1. Show that R[z]/ (x) ~ R.

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Q: The ring 5Z is isomorphic to the ring 6Z

A: Since the z is same for both and the multiplier is different i.e. 5 &6.

Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

Q: The ring Zs[i] has no proper ideals True O False

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Q: Let R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it…

A: Given that R is a commutative ring with identity.

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

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Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

Q: Suppose R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

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Q: Explain why the polynomial rings R[x] and C[x] are not isomorphic.

A: Suppose there exists an isomorphism φ:R[x]→C[x]. Because isomorphisms are by definition surjective,…

Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.

A: Solution

Q: Prove that R[x2, x3] is a subring of the ring R[x].

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Q: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…

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Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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Q: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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Q: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

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Q: Is the idcal (x² + 1, x + 3) C Z[x] a principal idcal? Explain. The ring Z[x]/(x² +1, x+3) is…

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Q: Show that the factor ring Q[r] (P(x)) has a zero-divisor

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Q: In the ring Z[x]/: 1- The inverse of x is:

A: The given ring is ℤ2x/x3+1. The objective is to find the inverse of x and to find Uℤ⊕ℤ.

Q: The ring Zs[i] has no proper ideals True False O O

A: We check whether Z8[I] has proper 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: b. Let R be a nontrivial ring such that, for each 0 + a E R there exists unique element x in R such…

A: Note: According to Bartleby guidelines; for more than one question asked, only the first one is to…

Q: Let R be a commutative ring such that a^2 = a for all a ∈ R, then show that a+a = 0.

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Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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Q: 1. Suppose that p: R → S is a ring homomorphism and that the image of o is not {0}. If R has a unity…

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Q: Let R be a ring and M be an R-module. Ther

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Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…