Let Z[i] be the ring of Gaussian Integers. Describe the quotient ring Z[i]/(2+i) up to isomorphism.

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 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: Show that the polynomial ring Z4 [x] over the ring Z₁ is infinite, but Z₁ [x] is of finite…

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

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Q: 12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only…

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Q: Let R[x] be the set of all polynomials over a ring (R ,+, . ). Then, (R[x] ,+ , .) form ring , which…

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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 with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

Q: If a ring R has characteristic zero, then R must have an infinite number of elements. true or false

A: We have find that the given statement  "If a ring R has characteristic zero, then R must have an…

Q: 1. Let f be an isomorphism of a ring R onto a ring R'. Show that (a) If R is an integral domain,…

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Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.

A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…

Q: Let T = Z - 5Z. Show that ZT is a local ring. What is its unique maximal ideal?

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Q: Prove that the only ideals of a field F are {0} and F itself.

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Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

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

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Q: Let R be a commutative ring with identity and I be ideal of R. Then I is primary if and only if…

A: The statement is true.

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

A: Solution:

Q: Give an example to show that the factor ring of a ring with zero divisors may be an integral domain.

A: (i)4ℤ8ℤ≈ℤ2 (ii) ℤ6ℤ

Q: Let R be a ring with a multiplicative identity 1R. Let u, an element of R, be a unit. Prove: u is…

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Q: The ring 5Z is isomorphic to the ring 6Z

A: Since the z is same for both and the multiplier is different i.e. 5 &6.

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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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: Show that each homomorphism from a field to a ring is either one to one or maps everything onto 0.

A: Let ϕ:F→R  be a ring homomorphism from the field F to ring R . Now, the  kernel of ϕ is ideal of F.…

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 = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

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Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R {a+ bV2|a,…

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Q: Let R′ be a commutative ring of characteristic 2. Show that                    I ={r∈R′|r^2 =0} is…

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Q: Let ø : F → R be a ring homomorphism where F is a field. Show that either o is one-to-one or ø is…

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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Show that there is a unique irreducible polynomial of degree 2 over F₂. Further, if a is a root of…

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Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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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: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R = {a+…

A: a) Let F be a field. We know that field has no proper ideals. The ideals of F are 0 and F only. The…

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: Let R be the set of all functions. It is known that (R,+,×) is a ring where + and × are both…

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Q: Let R be a ring with a multiplicative identity 1R. Let u, an element of R, be a unit. Prove: every…

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Q: Let A be a ring with identity e and let a, b E A such that ab = e. Then ba is a Idempotent element…

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Q: Is the idcal (x² + 1, x + 3) C Z[x] a principal idcal? Explain. The ring Z[x]/(x² +1, x+3) is…

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Q: 4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the…

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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: If R be a commutative ring with identity and a ∈ R is idempotent different from 0 or 1. Prove that R…

A: Given that R is a commutative ring  with unity. Also, a∈R  is an idempotent element ⇒a2=a ( a≠0,1)…

Q: 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: An element x in a ring is called an idempotent if x2 = x. Prove that  the characterstic of R is 0 or…

A: An element x in a ring is called an idempotent if x^2 = x

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: 1. Let f be an isomorphism of a ring R onto a ring R'. Show that (a) If R is an integral domain,…

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Q: Construct a non-trivial Cayley's table of a ring isomorphic to the ring Z,. Also write the mapping…

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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: 3. Let R be any commutative ring with unity, and let T[r] be the subset of all polynomials with zero…

A: Given that R is a commutative ring with unity, and T[x] be a the subset of all polynomials with zero…

Q: The ring Z-[i] has no proper ideals

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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…

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.