Let R be a commutative ring with unity. If I is a prime ideal of R prove that I[x] is a prime ideal of R[x]

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

A: We have to prove that factor ring R/I is commutative iff rs-sr is in R for all r and s in R.

Q: Let R be a commutative ring with unity, and let I be a proper idealwith the property that every…

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Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x € R:x" e A for some…

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Q: Let R be a commutative ring that does not have a unity. For a fixed a e R prove that the set (a) =…

A: Let R be a commutative ring that does not have a unity. For a fixeda∈ℝ, we need to prove that :…

Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x€ R:x e A for some positive…

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Q: Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only…

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Q: Let R be a ring with a subring S. Prove or disprove: If a ∈ R is nilpotent and a ∈ S, then a is also…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Let R be a commutative ring. If I and P are ideals of R with P prime such that I ¢ P, prove that the…

A: The ideal quotient of P and I is P:I=x∈R : xI⊂P which is again an ideal of R. Given that P is a…

Q: Indicate such a subring of the ring P[x] that contains P and is different from P but is not…

A: There are so many examples can be found.

Q: If R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.

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Q: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…

A: To prove that every prime ideal of R is maximal.

Q: Let I be an ideal of a commutative ring (R,+.)with unity, then I is no prime ideal iff RM is…

A: We know characterstic of an integral domain is either prime or zero. Let I be an ideal of a…

Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {xe R: x" e A for some…

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Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x€ R:x" e A for some…

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Q: (b) If M is a maximal ideal of a ring R then M is a prime ideal of R.

A: Given: If M is a maximal ideal of a ring R, then M is a prime ideal of R To prove or disprove the…

Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: Let R be a commutative ring with identity. Using the homomorphism theorem (Theorem 16.45) and…

A: Recall that in a ring A  not necessarily commutative and with an identity, an ideal M⊂A is a maximal…

Q: Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an…

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Q: 17. Let H and K be ideals of a ring R. Show that HNK is an ideal of R.

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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: Prove that a nonzero commutative ring with unity R is a field if and only if it has two ideals (0)…

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Q: Let R be a commutative ring with unity and let a ∈ R be fixed. Prove that the subset Ia = {x ∈ R :…

A: Given below the detailed solution

Q: Label each of the following statements as either true or false.   The only ideal of a ring R that…

A: Given Statement, The only ideal of a ring R that properly contains a maximal ideal is the ideal R.…

Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

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Q: Let R be a commutative ring. Prove that the principal ideal generated by the element x E R[x] is a…

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Q: Let f : R→ R' be a ring homomorphism of commutative rings R and R'. Show that if the ideal P is a…

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Q: a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that…

A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…

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: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: If A and B are ideals in a ring R such that A n B = {0}, prove that for every a E A and b E B, ab =…

A: Explanation of the answer is as follows

Q: Let A be a ring, and let I, P ⊂ A be ideals of A such that I ⊂ P. Show that the following are…

A: Use the relation between integral domain and prime ideal

Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: 1. Let R be a commutative ring with unity and let a e Rbe fixed. Prove that the subset Ia = {x E R:…

A: i have provided the detailed proof in next step

Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x e R: x" e A for some…

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Q: Let M be a commutative ring with identity x and R be a maximal ideal of M. Then prove that R is…

A: Prime Ideal : A prime ideal S of a commutative ring R is a proper ideal of R such that a, b∈R and…

Q: Let M be a proper ideal in a Boolean ring R with unity. Prove that (1) R/M is a Boolean ring and…

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Q: elements

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Q: Suppose that R and S are commutative rings with identities. Let Ø be a ring homomorphism from R onto…

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Q: Show such a subring of the ring P[x], which contains P and is different from P and is not isomorphic…

A: The given details: The ring is P[x]. To show that the subring of the ring P[x] which contains P and…

Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x e R:x" e A for some…

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Q: Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of…

A: Given: R is a commutative ring with unity and that I is an ideal of R. To prove: The set of all x∈R…

Q: Label each of the following statements as either true or false.   The only ideal of a ring R that…

A: The ring R is defined as a set that is not empty and contains a minimum of 1 element in it. This…

Q: Indicate such a subring of the ring P[x], which contains P and is different from P, but is not…

A: Image is attached with detailed solution.

Q: Let R be a commutative ring with unity  If I is a prime ideal of R, prove that I[x] is a prime…

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Q: 2) Let P + Q be maximal ideals in a ring R and a,b elements of R. Show that there exists c E R, such…

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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…

Q: 3. Let R be any commutative ring with unity, and let T[r] be the subset of all polynomials with zero…

A: Given that R is a commutative ring with unity, and T[x] be a the subset of all polynomials with zero…