This document defines and provides examples of rings. A ring is a set with two binary operations, addition and multiplication, that satisfy certain properties: 1) addition forms an abelian group, 2) multiplication is associative, and 3) the distributive laws hold. Simple examples of rings include the integers Z, rational numbers Q, real numbers R, and complex numbers C. The document also verifies that the set of even integers E is a ring and provides another example of a ring defined by addition and multiplication tables. Rings can be commutative or non-commutative and may or may not have an identity element.

1. WHAT IS A RING?
 A set of elements
 Closed under addition and multiplication
 Commutative group under addition
 Multiplication and Addition is associative
 Left and Right Distributive Laws hold
 Contains an identity element 0
 Contains additive inverses

2. ALTERNATIVE DEFINITION OF A
RING
A ring 𝑅 is a set defined by two operations (addition and
multiplication) and the following conditions hold:
1. 𝑅 forms an abelian group with respect to addition
2. 𝑅 is closed with respect to an associative multiplication
3. Two distributive laws hold in 𝑅

3. SIMPLE EXAMPLES OF RINGS
1. The set of ℤ of all integers
2. The set ℚ of all rational numbers
3. The set ℝ of all real numbers
4. The set ℂ of all complex numbers

4. EXAMPLE 2:
VERIFYTHATTHE SET 𝐸 OF ALL EVEN INTEGERS IS A RING
WITH RESPECTTO USUAL ADDITION AND MULTIPLICATION
IN ℤ.
Solution:
Properties Proof/Explanation
1. Closure If 𝑥 ∈ 𝐸 and 𝑦 ∈ 𝐸, then 𝑥 = 2𝑚 and 𝑦 = 2𝑛
with 𝑚 and 𝑛 in ℤ.
Addition: 𝑥 + 𝑦 = 2𝑚 + 2𝑛 = 2 𝑚 + 𝑛 ,
which is in 𝐸.
Multiplication: 𝑥𝑦 = 2𝑚 2𝑛 = 4𝑚𝑛 =
2(2𝑚𝑛), which is in 𝐸.
2. Associativity Addition and Multiplication in 𝐸 is associative.
(following the properties of integers since
even numbers contains in ℤ)
3. Commutativity Addition and Multiplication in 𝐸 is
commutative.

5. EXAMPLE 2:
VERIFYTHATTHE SET 𝐸 OF ALL EVEN INTEGERS IS A RING
WITH RESPECTTO USUAL ADDITION AND MULTIPLICATION
IN ℤ.
Solution:
Properties Proof/Explanation
4. Identity element 0 𝐸 contains the additive identity, since 0 + 2 =
2.
5. Distributive Laws The two distributive laws hold in 𝐸. (following
the properties of integers since even numbers
contains in ℤ)
6. Additive Inverse For any 𝑥 = 2𝑘 in 𝐸, the additive inverse of 𝑥
is in 𝐸, since −𝑥 = 2(−𝑘)

6. ANOTHER EXAMPLE OF A RING
The set 𝑆 = 𝑎, 𝑏 with addition and multiplication defined by the
tables:
+ 𝑎 𝑏
𝑎 𝑎 𝑏
𝑏 𝑏 𝑎
∗ 𝑎 𝑏
𝑎 𝑎 𝑎
𝑏 𝑎 𝑏

7. ANOTHER EXAMPLE OF A RING
The set T = 𝑎, 𝑏, 𝑐, 𝑑 with addition and multiplication defined by the
tables:
+ 𝑎 𝑏 𝑐 𝑑
𝑎 𝑎 𝑏 𝑐 𝑑
𝑏 𝑏 𝑎 𝑑 𝑐
𝑐 𝑐 𝑑 𝑎 𝑏
𝑑 𝑑 𝑐 𝑏 𝑎
+ 𝑎 𝑏 𝑐 𝑑
𝑎 𝑎 𝑎 𝑎 𝑎
𝑏 𝑎 𝑏 𝑎 𝑏
𝑐 𝑎 𝑐 𝑎 𝑐
𝑑 𝑎 𝑑 𝑎 𝑑

8. PROPERTIES OF RINGS
Every ring is an abelian additive group.
There exists a unique additive identity element 𝑧, (the zero of the ring)
Each element has a unique additive inverse, (the negative of the element)
The Cancellation Law for addition holds
− −𝑎 = 𝑎 & − 𝑎 + 𝑏 = −𝑎 + (−𝑏) for all 𝑎, 𝑏 of the ring
𝑎 ∙ 𝑧 = 𝑧 ∙ 𝑎 = 𝑧
𝑎 −𝑏 = − 𝑎𝑏 = (−𝑎)(𝑏)

9. TYPES OF RINGS
1. Commutative Ring – a ring for which multiplication is
commutative
2. Ring with Identity Element (Ring with Unity) – a ring having a
multiplicative identity element (unit element of unity)

10. EXAMPLES
1. ℤ, ℚ, ℝ, ℂ are all commutative rings with identity.
2. The set of real polynomials is also an example of a commutative ring
with an identity element.
3. The ring 𝐸 of all even integers is a commutative ring, but 𝐸 does not
have a unity.
4. The set of 2 × 2 matrices with 𝑅 =
𝑎 𝑏
𝑐 𝑑
𝑎, 𝑏, 𝑐, 𝑑 𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 is a
non-commutative ring with an identity element
1 0
0 1
.