Question 1A - Rings and subrings

Question 1

Definition : Ring

Answer 1

A ring (R, + , . , 0_R , 1_R) is a set R, with binary operations + , . : RxR -> R and distinguished elements 0_R, 1_R ϵ R such that :

1) (R, + , 0_R) is an abelian group
2) (R, . ,1_R) is a monoid
3) Multiplication distributes over addition:
x . (y+z) = x.y + x.z
(x+y) . z = x.z + y.z

Question 2

Definition : Abelian Group

Answer 2

Associative addition :

If x · y = y · x for all x, y ∈ G then we say that the group G is abelian.

(x+y) +z = x+ (y+z)

x+y = y+x

0_R + x = x = x+ 0_R

Question 3

Definition : Monoid

Answer 3

Associativity of multiplication :

  (x.y) . z = x .  (y.z)

   1_R . x = x = x . 1_R

Question 4

Simple rules (lecture 1)

Answer 4

1) x . 0 = 0 = 0 . x
2) (-1) . x = -x = x.(-1)
3) (x + y)^2 = x^2 + xy + yx + y^2
4) (-x)(-y) = xy

Question 5

Definition : Commutative Ring

Answer 5

A ring R is commutative if xy = yx for all x,y ∈ R

Question 6

Definition : Subring

Answer 6

Let R be a ring. Let S ⊆ R (S is a subset of the set R). S is a subring if S is a ring using +_R and ._R [in particular 0_R , 1_R ∈ S]. We write S ≤ R.

Question 7

Proposition : The subring test ON EXAM

Answer 7

Let R be a ring and S⊆ R, S is a subring iff
1) 1∈ S
AND
2) S is closed under subtraction and multiplication.

—————–Proof ——————–

If S is a subring then by definition 1 ∈S and S is closed under the ring operations.

Conversely assume 1 ∈S and S is closed under subtraction and multiplication.
Then 0 = 1−1 ∈S, and since S is closed under subtraction we know that (S,+)is a subgroup of (R,+) by the subgroup test.

Associativity and distributivity of
multiplication hold in S because they hold in R. #

Question 8

Definition : Divides

Answer 8

Given m,n ∈ Z(bar) we say m divides n, written m | n, if
n = qm for some q ∈ Z(bar)

Question 9

Definition : Congruence

Answer 9

Let x, y, n ∈ Z(bar) with n ≥ 0. We say x and y are congruent modulo n, written x ≡ y (mod n),
if n divides x − y

Question 10

Proposition : The principal of long division (pf not on exam)

Answer 10

Let x, n ∈ Z(bar) with n > 0.
Then there are unique integers q, r ∈ Z(bar) (the quotient and remainder) such that x = qn + r and 0 ≤ r < n.

Question 11

Proposition 2.4 r = r’ (pf not on exam)

Answer 11

Let x, y, n, ∈ Z(bar).
Write x = qn + r and y = q′n + r′ with
q, q′, r, r′ ∈ Z(bar) and 0 ≤ r, r′ < n.
Then x ≡ y (mod n) if and only if r = r′.

Question 12

Definition: Integral Domain

Answer 12

An integral domain is a commutative ring with no nonzero zero divisors.

Question 13

Definition : Congruence Class

Answer 13

The congruence class of x mod n, denoted [x]_n, or [x] for short if n is understood, is the set
[x] = {x + kn : k ∈ Z(bar)} = x + nZ(bar)
of all integers congruent to x mod n

Question 14

Definition : The ring of integers mod n

Answer 14

The ring of integers mod n, is the set
Z(bar)/nZ(bar) = {[0], [1], . . . , [n − 1]}
of congruence classes of integers mod n.
Addition and multiplication are defined by
[x] + [y] = [x + y], [x] · [y] = [x · y].
The additive and multiplicative identities are [0] and [1], respectively.