Let R be a ring. The center of R is the set {x E R | ax = xa for all a in R}. Prove that the center of a ring is a subring.

Q: Construct a homomorphism of rings p:Z[i] → Z,

A: Consider the rings ℤi and ℤ2. Define a map φ:ℤi→ℤ by φa+ib=0 ∀ a,b∈ℤ. Let a+ib, c+id∈ℤi.…

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: 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 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: The ring Z is isomorphic to the ring 3Z

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Q: Let R be a ring with identity 1 and let a be an element of R such that a2 = 1. Let S = { ara : r e…

A: we will use sub ring test.

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

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

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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: Let R=Z and R'= set of all even integers. Then %3D (R', +, *) is a ring, where a* b= ab V a, be R'.…

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Q: The ring Z is isomorphic to the ring 3Z O True False

A: Solution:

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 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: 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 ring with m elements. Show that the characteristic of Rdivides m.

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

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Q: Let a and b be elements of a ring R. Prove that the equation a+ x= b has a unique solution.

A: a and b are elements of ring R. We have to show that (a+x)=b has a unique solution.

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

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Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

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: Let R be a commutative unitary ring and let M be an R-module. For every r ERlet rM = {rx; x E M} and…

A: We shall solve first question only as you have asked more than one different question as per company…

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: 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: The rings Z and 5Z are isomorphic.

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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: 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 S be a homomorphism of rings, 1. If K is a subringof R, Is o(K) a subring of S? If so,…

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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: Let R be a commutative ring such that 2x=0 xe R. Then the mapping f: R →→R defined as f(x) = x² is a…

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Q: 3. Let R be a ring and b E R be a fixed element. Let and prove that T is a subring of R T = {rb | r…

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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 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: Is the mapping from Z5 to Z30 given by x → 6x a ring homomorphism? Note that the image of the unity…

A: In the given question we have to tell that " Is f(x)=6x is a ring homomorphism from (Z5,⊕5,⊗5) to…

Q: 16. Let f: R S be a ring homomorphism with J an ideal of S. Define I= {r ER| f(r) € J} and prove…

A: Given :  f : R →S be a ring homomorphism with J an ideal of S. To prove :  I = r∈R : f(r)∈J  is an…

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

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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: b. Let R be a nontrivial ring such that, for each 0 + a E R there exists unique element x in R such…

A: Note: According to Bartleby guidelines; for more than one question asked, only the first one is to…

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: Let R=Z and R'= set of all even integers. Then %3D (R', +, *) is a ring, where a*b = ab V a, be R'.…

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