a. Let R and S be commutative rings with unities and f:R -S be an epimorphism of rings. Prove that S is an integral domain if and only if kerf is a prime ideal of R.

Q: Let R be a ring and x be an indeterminate. Prove that the rings R[x]and R[x2] are ring-isomorphic.

A: Given That : Two rings R[x]  and R[x2]  To Prove: Two ringsR[x],  R[x2]  are ring-isomorphic

Q: Let R be a commutative ring and let a ∈ R . Show that I a = { x ∈ R ∣ a x = 0 } is an ideal of R.

A: Given: Let R be a commutative ring and let a ∈ R . To Show that      I a = {x∈R  ax = 0}  is an…

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

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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 R be a commutative ring with unit which is not an integral domain. Show that quotient and…

A: Question asks to show that if R is a commutative ring with a unit that is not an integral domain,…

Q: Let R be a ring such that R² = {0}. Prove that R is a division ring if and only if R has no…

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

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Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

A: The objective of the question is to determine whether the union of two ideals I and J of a ring R is…

Q: Let R be a commutative ring with unity and let a, b E R. Show that (a, b), the smallest ideal of R…

A: Here, R be a commutative ring with unity and let a,b∈R. It has to be proved that <a,b>, the…

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: Show that the union of a chain I, CI,c of ideals of a ring R is an ideal of R. (This exercise is…

A: Given: We are given a chain of ideals in a ring Objective: The objective is to show that the union…

Q: Let R, S be rings with unity and o : R→ S a ring homomorphism. Show that if o(1R) is a unit, then…

A: Given:- Let R,S be rings with unity ϕ:R→S a ring homomorphism. To Prove ϕ(1R)=1S

Q: Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.

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Q: Let R be a commutative ring with identity 1R ≠ 0R. Prove that R is an integral domain if and only if…

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

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

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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: Let R be a commutative ring with unity and let S≠R be an  ideal of R. Then R/S is an integral domain…

A: Integral domain and prime ideal.

Q: Let R be a ring. An element a E R is called nilpotent if there exists an integer n> 0 such that an…

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Q: Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is…

A: Given R be a commutative ring with unit element. If f(x) is a prime ideal of R[x] then we have to…

Q: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.

A: We are given that R be a commutative ring with unity. We have to show that  R is a field if and only…