a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be written as finite sums of elements of the form xy, where x E U, y E V. Prove that UV is an ideal of R.

Q: Prove that the rings Z3[x]/ and Z3[x]/ are isomorphic.

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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: 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: Prove that the only homomorphisms from Z to Z (Z being the ring of integers) are the identity and…

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

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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: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

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: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I=I ∩ rad R

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Q: Prove that the rings Z3[x]/ and Z3[x]/ are isomorphic

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: Let 1={0,2} CZ. Prove if ring Z,/l is isomorphic with a ring Z,

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

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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…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

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: element a ∈ R, define the Annihilator of a denoted as Ann(a), as Ann(a) = {r ∈ R| r.a = 0} Show that…

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Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

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: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J…

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Q: given an example for a ring R has ideal I such that : VI #I + IC VI

A: Note:- If you don't understand anything or want anything else in this problem then please resubmit…

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: 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: (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: 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 R be a ring with identity. If ab and a are units in R, prove that b is a unit.

A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

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: 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: 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 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: 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: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).

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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: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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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: 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 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: 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…

Q: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…

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

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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: 5. Let A and B be two ideals of a commutative ring R with unity such that A +B = R. Show, that AB=A…

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