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

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

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Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

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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: if A and B are ideals in a ring R such that A intersect B ={0}, prove that for every a in A and b in…

A: Let A and B are ideals of a ring R such that A∩B=0

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: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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Q: Let R be a ring of all real numbers , Show that H= { m+nV2 | m, ň E Z} is a subring of R.

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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: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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

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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 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 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 Roll no be a ring with ideals I and J , such that I ⊆ J . Then J/I is an ideal of Roll no/I .

A: Ideal: A non-empty subset I of  a ring R is called ideal in a ring R if following conditions holds:…

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: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: Let I and J be ideals of a ring R. Prove or disprove (by counterexample) that the following are…

A: Given that I and J are two ideals of a ring R Ideal Test: A nonempty subset A of a ring R is an…

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: 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: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…

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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: Given that I = {(a,b) E R | a and b are even numbers} is an ideal of a ring R = Z × Z, construct…

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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: 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: 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 and let I be a proper ideal of R. Prove that R/I is a…

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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: Let R be a ring. Prove that the set S = x R / xa = ax, a Ris a subring of R .

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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: 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: 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: Let M and N be ideals of a ring R and let H = {m+n | m∈ M, n ∈ N} (a) Show that H is an ideal of R.…

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

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

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Q: If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.

A: R1 and R2 are subrings of the ring R, prove that R1∩R2 is a subring of R

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 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: Let I= {(a, 0)| a eZ}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z. Id if…

A: We knew that an ideal I of a ring R is said to be prime ideal if for a ,b ∈ R and ab ∈ I this imply…

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: 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: Prove that Q[x]/<x2 - 2> is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

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

A: Solution :

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…