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 I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.

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Q: Let R be a commutative ring. Then R is an integral domain if and only if ab= ac implies that b = c,…

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Q: Let ø : R  → R’  be a ring homomorphism and let N be an ideal of R.  Let N’ be an ideal either of ø…

A: Ideal: A non-empty subset I of a ring R is said to be ideal in a ring I if it satiesfies following…

Q: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.

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

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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: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then... ..... .... (a) 1 = R (b)…

A: {R, +, .) is a ring and it is given that I is not a maximal ideal in R.From this, we conclude that I…

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

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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: Suppose A is a Noetherian unital commutative ring and I is an ideal of A. Prove that A/I is…

A: Some basic results of module theory required to understand the following proof better . If R is a…

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: 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 R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

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: 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 ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…

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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 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 an integral domain. Prove that {0R} is a prime ideal.  Let R be a ring and let p ∈ R be…

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Q: 4. Let p: R S be a ring homomorphism. Show that J = ker p is a prime ideal if S is a domain. Show…

A: Fundamental theorem of homomorphism: Let R and S be rings. Consider the homomorphism φ:R →S. Then,…

Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the overlying and M⊆R is maximal ideal, then…

A: Given that  ϕ:R → S be a ring homomorphism. This implies that R and S are commutative rings with 1.…

Q: For a fixed element a of a ring R, prove that the set {x ϵ R I  ax = O} is a subring of R.

A: Given :            The ring R and a fixed element 'a' of R.   To prove that the set x ∈ R | ax =…

Q: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: Let R be a commutative ring with unity, and let c ER be a fixed element. (a) Prove that the set A =…

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

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Q: a) Let Ø: R – S be a homomorphism of rings R and S and let B be an ideal of S. Show that 01(B) =…

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Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D

A: The term radical is used when we think about ideals and when we talk about ideals definitely…

Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

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Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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Q: 2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime…

A: Given R be a commutative ring with unity and if I is a prime ideal of R. Then we have to prove that…

Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the covering and M⊆R is maximal ideal, then…

A: Given φ:R→S be a ring homomorphism. Let,, φ is covering and M⊂R is maximal ideal.  To prove that…

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

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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: 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: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

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

A: Let R be a commutative ring with unity. If I is a prime ideal of R we have to prove that I[x] is a…

Q: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

Q: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…

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: 18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root…

A: Given: Let ϕ:ℂ→ℂ be an isomorphism or rings such that ϕ(a)=a for each a∈ℚ. Suppose r∈ℂ is the root…

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

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Q: Let A = {S : S is a subring of C and e S}, and let R = N S be SEF the intersection of all these…

A: Given that A=S| S is a subring of ℂ and 12∈S and let R=∩S∈AS. (a) Consider the ring M=3k: k∈ℤ.…

Q: Let R be a commutative unital ring. Let I be an ideal of R. Denote by (I) the ideal in R[a]…

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

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Q: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].

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