Section 4 - Rings

Question 1

Binary Operation

Answer 1

A binary operation on a set A is a function *: A x A -> A. Usually we write a * b

Question 2

Ring

Answer 2

A ring is a non-empty set R together with two binary operations: addition and multiplication. It satisfies all of the following: axioms of addition, associativity of multiplication, distributivity

Question 3

Axioms of addition

Answer 3

(A1) For all a,b,c∈R, (a + b) + c = a + (b + c) (associativity of addition)
(A2) For all a,b∈R, a + b = b + a (commutativity of addition)
(A3) There exists 0∈R such that, for all a∈R, 0 + a = a (existence of a neutral element for addition)
(A4) For all a∈R, there exists b∈R such that b + a = 0 (existence of additive inverses)

Question 4

Associativity of Multiplication

Answer 4

For all a,b,c∈R, (ab) × c = a × (bc)

Question 5

Distributivity

Answer 5

For all a,b,c∈R:
(i) a(b+c) = ab + ac
and
(ii) (b+c)a = ba + ca

Question 6

Commutative Rings

Answer 6

A ring R is called commutative if its multiplication operation is commutative: ab = ba for all a,b∈R

Question 7

Unital Rings

Answer 7

A ring R is called unital if there is an element 1∈R such that, for all a∈R: 1 x a = a x 1 = a

Question 8

Units

Answer 8

Let R be a unital ring. An element a∈R is called a unit if there are b,c∈R such that ba=ac=1

Question 9

Z/nZ

Answer 9

[a] = {b∈Z | b ≡ a mod n} = {b∈Z | n divides b - a}

Question 10

Units in Z/nZ

Answer 10

Let n be a positive integer and a∈Z. Then a is a unit of Z/nZ iff gcd(a, n) = 1

Question 11

Subring

Answer 11

Let R be a ring. A subring of R is a subset of R that is itself a ring with respect to the same operations of addition and multiplication as in R.

Question 12

Properties of Subrings

Answer 12

Let R be a ring and S a subset of R. Then S is a subring of R iff all of the following hold:
(i) S is non-empty
(ii) ∀a,b∈S, a-b∈S (closure under subtraction)
(iii) ∀a,b∈S, ab∈S (closure under multiplication)

Question 13

Ideal

Answer 13

Let R be a ring. An ideal of R is a nxn non-empty subset I of R such that both of the following hold:
(i) ∀a,b∈I, a-b∈I (closure under subtraction)
(ii) ∀a∈I and ∀r∈R, ra∈I and ar∈I (closure under multiplication by ring elements)

Question 14

Concerning ideals and subrings, fill in the blanks: Every ______ is a __________

Answer 14

Every ideal is a subring

Question 15

Principal Ideal

Answer 15

An ideal that consists of only 1 element

Question 16

Ideal generated by X

Answer 16

The intersection of all ideals of R containing X

Question 17

Quotient Ring (R/I)

Answer 17

Let R be a ring and I an ideal of R. R/I = {a+I | a∈R}. R/I is a ring called the quotient of R by I.

Question 18

Coset

Answer 18

The equivalence class a+I is called a coset of I containing a

Question 19

What condition can be used to determine if a+I=b+I for some a,b∈R?

Answer 19

If a,b∈R, then a+I=b+I iff a-b∈I