Question

In: Advanced Math

(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}

Solutions

Expert Solution

Colby Messinger answered 1 year ago
Next > < Previous

Related Solutions

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,...
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.
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.
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.
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?
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.
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 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 t be a positive integer. Prove that, if there exists a Steiner triple system of...
Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT