Refresher - Groups, Rings, Fields

The properties that we just mastered – closure, commutative, associative, distributive, identity, inverse – together with a set of elements and an operation, form the basis of abstract algebra.

Abstract algebra focuses on the concepts of groups, rings, and fields.  These concepts provide the structures for the study of modern algebra. The animation on the right shows conceptually, in order, groups, then rings, then fields.

The properties for a group give us the “rules” we need to work with solving equations. Mathematically speaking, a group, G, is a set of elements with one operation, such as addition or multiplication, ${\displaystyle a•\mathrm{b.}}$

 ${\displaystyle •}$ refers to an operation, not necessarily multiplication. We will also begin to use mathematical notation, for example
${\displaystyle \mathrm{a }\in S}$ means that a is an element of set S.

A group would be a set S and an operation ${\displaystyle •}$ that satisfies all of the following properties:

1. If ${\displaystyle a,\mathrm{b }\in S}$, then ${\displaystyle a•b}$ is also ${\displaystyle \in S}$ (Closure)
2. ${\displaystyle \mathrm{a }• (\mathrm{b }• c)=(\mathrm{a }• b) • c}$ for all ${\displaystyle a,b,\mathrm{c }\in S}$ (Associative Property)
3. There is an element ${\displaystyle \mathrm{e }\in S}$ such that for all
   ${\displaystyle \mathrm{a }\in S,\mathrm{a }• e=\mathrm{e }•\mathrm{a }=a}$ (Identity Element)
4. For each ${\displaystyle \mathrm{a }\in S}$, there is an element ${\displaystyle \mathrm{b }\in S}$ such that ${\displaystyle \mathrm{a }• \mathrm{b }= \mathrm{b }• \mathrm{a }= e}$, where e is the identity (Inverse Element)

Example 1

In chemistry, we can look at mixing liquids and determining pH's as a group. In this group the pH is always between 0 and 14, and a pH of 7, like pure water, is the identity. 

If we add liquid A with a pH of x to liquid B with a pH of y, the resulting liquid (A+B) will have a pH of z.   [Closure]

If we add liquid A with a pH of x to liquid B with a pH of y then add liquid C with pH of w [(A + B) + C], we would get the same pH as when we add liquid A to the mixture of liquid B and liquid C  [A + (B + C)]. The resulting liquid will have a pH of g.   [Associative Property]

If pure water is added to liquid A or liquid B, the pH is not changed. [Identity]

If we take a base of a certain pH there exists an inverse that can turn the mixture into a pH of 7 and neutralize the base.  [Inverse]

Example 2

A military clock, numbering 0-23 as a modular 24 ,would be an example of a group. The operation would be moving the clock forward.

At 0800 hour (8 AM), after 9 hours elapsed, we are at 1700 hour (5 PM). [Closure]

In another nine hours, we are at 0200 hour (2 AM).  (8 + 9) + 9 = 8 + (9 + 9) [Associative Property]

If we move the clock forward 0 hours, we have our identity element.  [Identity]

Inverses exist (the inverse of 10 is 14), since at 1000 hour 14 more hours gets us back to 0). [Inverse]

 A group is said to be abelian Links to an external site. if ${\displaystyle x\cdot y=y\cdot x}$ for every ${\displaystyle x,y\in G}$ [Commutative Property].

Typical examples of group operation include addition and multiplication. Subtraction and division are usually not group operations since they are not associative.

Matrices Links to an external site. form a group under addition, multiplication, and transposition. 

Some examples of sets of numbers that form a group under the operation addition include integers (Z), rational numbers (Q), real numbers (R ), complex numbers (C ), vectors, and matrices. Groups under multiplication include Q, R, and C. 

Mathematically speaking, a ring consists of a set, such as the real numbers, with two operations (for example, denoted addition and multiplication), so that the set with the operation of addition forms an Abelian group, the operation of multiplication is associative (but not necessarily commutative), and the distributive property holds -- ${\displaystyle a(b+c)=ab+ac}$ and ${\displaystyle (a+b)c=ac+b\mathrm{c.}}$

If the operation of multiplication is commutative, we call the ring a commutative ring.

Examples of rings include integers, polynomials, and matrices. The basic commutative rings in mathematics include the integers, Z, the rational numbers, Q, the real numbers, R, the complex numbers, C, and polynomials.

Note:
Matrices are a good example of a non-commutative ring because, in general, matrix multiplication is not commutative.

Mathematically speaking, a field (Links to an external site.), F, is a set with two operations, addition and multiplication, such that the set with the operation of addition forms an Abelian group, and the set with the operation of multiplication also forms an Abelian group.  Fields are commutative rings which have no divisors of zero. A divisor of zero would be the existence of two elements, neither of which was zero, but whose product was zero.

For example, in modular or clock arithmetic, we have a set of elements and an operation. In modular 4, we have the elements 0, 1, 2 and 3. This would not meet the requirements of non-zero divisors, because in this system 2*2=0.

Essential field properties include: closure, associativity, existence of an identity element, existence of inverses, and commutativity for the operations of addition and multiplication. The identity elements for addition and multiplication must be different elements. The property of distributivity describes how the two operations interact with each other. In a field we can perform the operations addition, multiplication and because of additive and multiplicative inverses, we can also perform subtraction and division.

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