Thread: Rings: Don't quite understanding the basics...

Hi,
I can't quite get my head round rings, fields etc.

I can understand that for a ring, for example, the elements have to obey fundamental laws such as associate/commutative addition & multiplication, inverses etc, and this makes sense to me when I think of a set of integers modulo n, for example.

In a book that I'm reading it states ... that the underlying set of a ring need not be a set of numbers, and then it says we might take any set 'T' and let S = {all subsets of T}.
then define, for a & b which are members of S:
a + b = (a intersection b) - (a union b), and
a.b = a union b
So, I'm starting to see that the 'meaning' of '+' and '-' can be applied to a set ... I think? and I'm having trouble quite grasping what is going on here. It's like the fundamental laws are being applied to sets, and we can also define what the operations '+' and '-' mean when used in a Set context.

I'm just not quite getting the bigger picture here (not necessarily with regard to the example above, just in general). Can someone help me to understand please.

Many Thanks.

JG.

oops, should have said ...

a + b = (a union b) - (a intersection b), and
a.b = a intersection b

Maybe it doesn't make any difference though if you can define them to be what you wish?

Originally Posted by JGarner

Hi,
In a book that I'm reading it states ... that the underlying set of a ring need not be a set of numbers, and then it says we might take any set 'T' and let S = {all subsets of T}.
then define, for a & b which are members of S:
a + b = (a intersection b) - (a union b), and a.b = a union b

Originally Posted by JGarner

oops, should have said ...
a + b = (a union b) - (a intersection b), and
a.b = a intersection b
Maybe it doesn't make any difference though if you can define them to be what you wish?

Indeed it makes quite a difference!
a + b = (a intersection b) - (a union b) equals the empty set for every a & b.

The point of this is to have you verify that all the ring properties are satisfied using this definition.
You check each one of them.
For example, can you show that the empty set is the additive identity?
Is addition commutative?
Is there an identity for multiplication?
ETC.

Originally Posted by JGarner

Maybe it doesn't make any difference though if you can define them to be what you wish?

You can define whatever operations you want, as long as it satisfies the Ring Axioms. (Read the begining, later on it is more complicated).

sorry for all the questions ...

So what we're saying is that the '+' operator doesn't actually mean 'addition' in the standard conventional sense?
It is being used in this context simply to represent an 'operation' between two operands?
In this example therefore we're saying that by defining the operations '+' and '.' in the manner above for Sets then the Ring Axioms are met?
Therefore for any 'objects' if we can define two operations '+' and '.' and the ring Axioms are met, then we have a Ring?

What can these other 'objects' be apart from numbers and sets?

How do we tie in all of the above to the associative law for example?
If a + b = (a intersection b) - (a union b),
then according to the associative law, (a + b) + c = a + (b + c), but we can't split '(a intersection b) - (a union b)' into separate terms to allow us to add a 'c' term???
Or maybe we'd have to say:
Let a + b = 'X',
then a + b + c = 'X' + c = (X intersection c) - (X union b) ...., hmmm, not sure how we could then show associative law from this ....????

I'm still confused.

Originally Posted by JGarner

sorry for all the questions ...

So what we're saying is that the '+' operator doesn't actually mean 'addition' in the standard conventional sense?
It is being used in this context simply to represent an 'operation' between two operands?
In this example therefore we're saying that by defining the operations '+' and '.' in the manner above for Sets then the Ring Axioms are met?
Therefore for any 'objects' if we can define two operations '+' and '.' and the ring Axioms are met, then we have a Ring?

Correct. You just say + we shall define to be this and this and this .... (but this one has to be commutative).

And then you say * we shall define to be this and this and this .... (and show thou satisfy the axioms).

What can these other 'objects' be apart from numbers and sets?

Call them "elements" the term "objects" is used for something else.
Yes, they do not need to be numbers.


Do you know what "group" is? If you do, same idea, only more general.

Originally Posted by JGarner

How do we tie in all of the above to the associative law for example? If a + b = (a intersection b) - (a union b),

Once again it is a + b = (aUb) - (a ^ b). It does make a different in the way the additive operator is defined. The ring of subsets is a very important concept in measure theory and integration. In set theory this operation is know as the symmetric difference of two sets. Proving the associative law is difficult because it is so messy in notation. But all the other ring properties are quite easy to show.