(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b)...

(3) Let m be a positive integer. (a) Prove that Z/mZ is a commutative ring. (b) Prove that if m is composite, then Z/mZ is not a field.

(4) Let m be an odd positive integer. Prove that every integer is congruent modulo m to exactly one element in the set of even integers {0, 2, 4, 6, , . . . , 2m− 2}

Expert Solution

Colby Messinger answered 2 years ago

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and...

Prove (Z/mZ)/(nZ/mZ) is isomorphic to Z/nZ where n and m are integers greater than 1 and n divides m.

8.Let a and b be integers and d a positive integer. (a) Prove that if d...

8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

7. Let m be a fixed positive integer. (a) Prove that no two among the integers...

7. Let m be a fixed positive integer. (a) Prove that no two among the integers 0, 1, 2, . . . , m − 1 are congruent to each other modulo m. (b) Prove that every integer is congruent modulo m to one of 0, 1, 2, . . . , m − 1.

Consider the group Z/mZ ⊕ Z/nZ, for any positive integers m and n that are divisible...

Consider the group Z/mZ ⊕ Z/nZ, for any positive integers m and n that are divisible by 4. How many elements of order 4 does G have, and why?

Prove the following: (a) Let A be a ring and B be a field. Let f...

Prove the following: (a) Let A be a ring and B be a field. Let f : A → B be a surjective homomorphism from A to B. Then ker(f) is a maximal ideal. (b) If A/J is a field, then J is a maximal ideal.

let d be a positive integer. Prove that Q[sqrt d] = {a + b sqrt d|...

let d be a positive integer. Prove that Q[sqrt d] = {a + b sqrt d| a, b is in Q} is a field. provide explanations.

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Given a commutative ring with identity R. A. Prove that a unit is not a zero...

Given a commutative ring with identity R. A. Prove that a unit is not a zero divisor. B. Prove that a zero divisor cannot be a unit.

Let A = Z and let a, b ∈ A. Prove if the following binary operations...

Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity). (a) (3 points) f(a, b) = a + b + 1 (b) (3 points) f(a, b) = a(b + 1) (c) (3 points) f(a, b) = x2 + xy + y2

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