If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the characteristic of R.

Q: Show that a unit of a ring divides every element of the ring.

A: Let us assume a unit in a ring R as a. Let us assume any element as x in that ring R.  By using the…

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…

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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 commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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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: Find an example of a commutative ring A that contains a subset, say S, such that for every s E S we…

A: We can find n commutative ring A such that A contains a subset S such that for every s in S we have…

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

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Q: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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Q: Indicate such a subring of the ring P[x] that contains P and is different from P but is not…

A: There are so many examples can be found.

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

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Q: If R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.

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Q: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

Q: Let R be a commutative ring and a e R. Show that if a is nilpotent, then ab is nilpotent for each be…

A: Solution. Since a is nilpotent , an = 0 for some n ∈ N.

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: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…

A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.

Q: For any element a in a ring R, define (a) to be the smallest ideal of R that contains a. If R is a…

A: An ideal is a non-empty sub set I of a ring R, such that

Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

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Q: What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S:…

A: What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S:…

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 R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it…

A: Given that R is a commutative ring with identity.

Q: Is the ring Z/6Z and Z6 are isomorphic.

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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: b) Prove that, if S is a ring with characteristic 0, then S infinite.

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Q: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.

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Q: If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative…

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Q: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

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: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…

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Q: The rings Z and 5Z are isomorphic.

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Q: Show if R is a commutative ring then R[x] is also a commutative ring.

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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: Describe all ideals of the following quotient ring: Q[x]/(x² – 5x + 6)

A: Given:Q[x](x2-5x+6)

Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

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Q: Let a and b be elements in a ring R. If ab is a zero divisor, prove that either a or b is a zero…

A: Given :          ab is a zero divisor where a and b are elements in ring R. Formula :        Zero…

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 ring such that a^2 = a for all a ∈ R, then show that a+a = 0.

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Q: Show such a subring of the ring P[x], which contains P and is different from P and is not isomorphic…

A: The given details: The ring is P[x]. To show that the subring of the ring P[x] which contains P and…

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. 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 be a ring and M be an R-module. Ther

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Q: Indicate such a subring of the ring P[x], which contains P and is different from P, but is not…

A: Image is attached with detailed 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…

Q: The number of zero divisors of the ring Z, O Zg is

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Q: Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a…

A: Let R be a ring with unity e, verify the mapping θ:Z→R defined by θx=x.e is a homomorphism If R and…

Q: The ring Zp2, has exactly-----------maximal ideals

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