MAKING SENSE OF MATHEMATICAL RINGS AND FIELDS

https://www3.imsa.edu/programs/e2k/curriculum/68reflections.mod/teaching

Last night was a very productive night. As I went over my old math notes, I came across a lecture on rings. This has been a lecture I'd never understood completely. As I read on, I begun to understand what a ring was. First I understood that for a set to be a ring, certain math properties must apply withing the set. But it all was so confusing until I begun to analyze more carefully. I ask to myself what is this about and how can I understand it? Let's take a look at a most original set of numbers: the set of integers:

..........-1, 0, 1, 2, ............

I found out that this set is a commutative ring and by definition an integral domain as it does not contain any zero divisors. zero divisors came into existence in the study of modular mathematics.


An example of a zero divisor would be 2•2 = 0. Here 2 is a zero divisor as 0 divided by 2 will give us 2 in the modular set : (0,1,2,3) this is called the set of all integers mod4. To give an example if we multiply 2•0 we get 0. If we multiply 2•1 =2 ; 2•2 = 4 but there is no four in this set so 4 becomes 0. Similarly 2•3 = 6 but there is no 6 so 2. Therefore clearly 2 is a zero divisor in the sense that 2 is not zero and yet when multiplying 2•2 we get 0. This makes 2 a zero divisor. Of importance is the fact that in the set of integers there are no zero divisors. The only way we can get a•b = 0 is if a or b is zero. This makes the set of all integer an integral domain. An integral domain is a ring. In particular the set of integers is a commutative ring. The fact that the set of integers has no inverse element such that a• -a = 1 deprives the integers from being a mathematical field unlike the field of the rational numbers where 2/5 • 5/2 = 1. The inverse element in this case is the reciprocal fraction, something clearly missing in in integers which are void of fractions. Thus, finally I got to understand fields and rings.

WHAT IS A COMMUTATIVE RING?

THE SET MUST BE A COMMUTATIVE GROUP: That is the following properties for addition must apply:

CLOSURE PROPERTY OF ADDITION (Magma or groupoid)
1) The result of adding any two members of the set, must give rise to another member in the set. For example: in the set of all integers (a ring) adding 3 and 4 give 7. Clearly, 3, 4, and 7 are integers. Thus adding any two integers will give rise to another integer, i.e; the sum of any a and b is in the set of all the integers. This property is called the closure property. Let a and b be in the set then a+b is in the set.

ASSOCIATIVE PROPERTY OF ADDITION (Semigroup)
2) letting a, b and c be in the set will ensure that (a+b)+c =a+(b+c), that is the associative property applies. (This makes it a semi-group)

IDENTITY PROPERTY OF ADDITION (Monoid)
3) Given any a there exist an element 0 such that a+0 =a. It must be true that in the set you must find a zero element. Clearly we can find the zero in the set of the integers as by definition the set of the integers are the positive and negative natural numbers including zero.

INVERSE PROPERTY OF ADDITION (group)
4) This property says that given any a in the set, there must also exist an -a, and this is clearly true in the case of the integers as every positive integers has its own corresponding opposite negative integer. Example: 4+( -4) = 0

COMMUTATIVE PROPERTY OF ADDITION (Abelian Group)
5) Given any a and b we have that a+b = b+a and this is the commutative property. that is given any two integers we can commute them and get the same answer.



THE SET MUST BE A COMMUTATIVE UNDER MULTIPLICATION TO A COMMUTATIVE RING, OTHER WISE IT IS SIMPLY A RING. The following properties must apply:

CLOSURE PROPERTY OF MULTIPLICATION
1) Given any elements a and b then a•b must be an element in the set. Clearly given any two integers their product is also an integer. Example: 2•3 = 6.

ASSOCIATIVE PROPERTY OF MULTIPLICATION (Seudoring)
2) Given any elements in the set a, b and c; the associative property of multiplication must apply. This clearly is the case in the set of the integers. That is, (a•b)c = a(b •c) This is clearly true when multiplying any three integers. Example: (3•2)•5 = 3•(2•5) = 30.

IDENTITY PROPERTY OF MULTIPLICATION (Ring)
3) Given any element a in the set, a•1 =a = 1•a

COMMUTATIVE OF MULTIPLICATION (Commutative Ring)
4) Given any elements in the set a and b then a•b = b•a: Example -2• 3 = 3• -2 = -6

ZERO PROPERTY OF MULTIPLICATION (Integral Domain)
5) Given any a and b, a•b = 0 or b•a = 0 if and only if a = 0 or b = 0.

INVERSE PROPERTY OF MULTIPLICATION (Field)
6) For any a then a• -a = 1 = -a• a. (This is the definition for a ring to be a field.)

So you see these is pretty neat stuff considering that most people in the world have no idea whatsoever about abstract algebra. But as I always said: Understanding is directly proportional to the way in which material is presented to the world. Seeing what is possible and not possible in the world makes way lot of sense when we consider that not all teachers are philosophers nor all philosopher teachers.