(Q,+, .) is a commutative ring : confusion

[bentley4]

A statement in a book of analysis I have says:
(Q,+, .) is a commutative ring with an identity element.

I assume by its notation (Q,+,.) is a tuple.(correct?)
There are several questions that come to mind:
1. Why is there an order between Q,+ and . ?
2. As far as I know, a tuple is an element of a cartesian product of sets. A ring $\subset$ set. So in other words the book says that the tuple (Q,+,.) is a set. How can a tuple be a set? Isn't that like saying that an element is a set?
3. Of what is + an element of? The set of numeral operators? If so, why can't I find it on the web?

 

[micromass]

A statement in a book of analysis I have says:
(Q,+, .) is a commutative ring with an identity element.

I assume by its notation (Q,+,.) is a tuple.(correct?)
There are several questions that come to mind:
1. Why is there an order between Q,+ and . ?

To be able to distinguish them. Q is the set of all the elements and + and . are the operations on the ring.

2. As far as I know, a tuple is an element of a cartesian product of sets. A ring $\subset$ set. So in other words the book says that the tuple (Q,+,.) is a set. How can a tuple be a set? Isn't that like saying that an element is a set?

In mathematics, everything is a set. Even tuples are sets. For example:

$$(a,b):=\{\{a\},\{a,b\}\}$$

By very definition of (a,b). So we see that (a,b) is defined as a set. Now, we also define

$$(a,b,c):=(a,(b,c))$$

So it is defined as two pairs. This is again a set. So we see that a tuple is a set.

3. Of what is + an element of? The set of numeral operators? If so, why can't I find it on the web?

+ is a binary operation. It is a function

$$+:\mathbb{Q}\times \mathbb{Q}\rightarrow \mathbb{Q}$$

 

[I like Serena]

Hi bentley4!

I suspect the statement in your book meant that Q is a set and the (Q,+,.) is a ring on that set.
A ring is defined by a set combined with an additive operator and a multiplicative operator, making it a 3-tuple.
We don't usually say that a ring is a set, although it is possible of course to talk about a set of which the elements are rings as micro explained.

Note that often people talk about the ring Q, although actually (Q,+,.) is meant.

Btw, you can not change the ordering without consequences.
The + and . have different definitions and cannot be exchanged.

 

[AlephZero]

I think you are trying to read too much into this. Just interpret "(Q,+,.)" as a shorthand for

"The set of rational numbers, an additive operator which is the standard addition operation on rationals (i.e. the way you learned to add fractions in elementary school), and a multiplicative operator which is the conventional multiplication operation on rationals, form a commutative ring with an identity element."

But life's too short to keep writing out all that, so mathematicians don't.

 

[bentley4]

To be able to distinguish them. Q is the set of all the elements and + and . are the operations on the ring.

You can just as well distinguish elements by using a set. I still don't understand why they have to be ordered. What do you mean with distinguish, just declare them? Everything has to be declared that is assumed that it exists, I agree. But why tuples?

In mathematics, everything is a set.

K. So than each element is a set on its own as well. 2 or more sets have the property that you can use the following operations on it: union, intersection, complement and cartesian product. So we can also perform these operations on 2 or more elements?

+ is a binary operation. It is a function
$$+:\mathbb{Q}\times \mathbb{Q}\rightarrow \mathbb{Q}$$

Hmm, k. I don't completely understand this yet but that's ok. I haven't treated functions rigourously and I'll come back to this if I still don't understand it after studying functions on my own.

 

[bentley4]

Hi bentley4!
Btw, you can not change the ordering without consequences.
The + and . have different definitions and cannot be exchanged.

Hey 'I like Serena',

Could you show how the ordering matters?
You say the ordering of + and . matters. So you imply that the ordering of Q doesn't matter?

 

[I like Serena]

Hey 'I like Serena',

Could you show how the ordering matters?
You say the ordering of + and . matters. So you imply that the ordering of Q doesn't matter?

In the 3-tuple (Q,+,.), Q has to be a set, and both + and . have to be binary operators that operate on 2 elements of the set Q that yield an element from Q.

Furthermore, each element q in Q has to have an additive inverse (denoted -q).

However, for a ring, it is not required that each element q has a multiplicative inverse (denoted q^-1) and indeed 0 does not have a multiplicative inverse anyway.

 

[micromass]

You can just as well distinguish elements by using a set. I still don't understand why they have to be ordered. What do you mean with distinguish, just declare them? Everything has to be declared that is assumed that it exists, I agree. But why tuples?

Because the ordering matters. A ring is by definition a tuple (R,+,.) where R is the set of elements, where + is the addition and where . is the multiplication. You can't write (+,Q,.) because that would imply that + is the set of elements and that Q is the addition. This is senseless.

Also, we can't just write {Q,+,.}, since we would have no idea then what the set of elements is, what the addition is and what the multiplication is. Indeed, {Q,+,.}={+,Q,.}=...

K. So than each element is a set on its own as well. 2 or more sets have the property that you can use the following operations on it: union, intersection, complement and cartesian product. So we can also perform these operations on 2 or more elements?

Yes, you can perform those operations on elements. But it is useless to do so, that's why you have never seen that before.

Hmm, k. I don't completely understand this yet but that's ok. I haven't treated functions rigourously and I'll come back to this if I still don't understand it after studying functions on my own.

 

[bentley4]

life's too short to keep writing out all that, so mathematicians don't.

True but without consistency in notation and semantics accuracy is lost eventually. Than we need to use new symbols or existing symbols that are infrequently used in that context.

 

[bentley4]

In the 3-tuple (Q,+,.), Q has to be a set, and both + and . have to be binary operators that operate on 2 elements of the set Q that yield an element from Q.

Furthermore, each element q in Q has to have an additive inverse (denoted -q).

However, for a ring, it is not required that each element q has a multiplicative inverse (denoted q^-1) and indeed 0 does not have a multiplicative inverse anyway.

K, thanks. I think I need to understand operations better.

 

[bentley4]

Because the ordering matters. A ring is by definition a tuple (R,+,.) where R is the set of elements, where + is the addition and where . is the multiplication. You can't write (+,Q,.) because that would imply that + is the set of elements and that Q is the addition. This is senseless.

Also, we can't just write {Q,+,.}, since we would have no idea then what the set of elements is, what the addition is and what the multiplication is. Indeed, {Q,+,.}={+,Q,.}=...




Yes, you can perform those operations on elements. But it is useless to do so, that's why you have never seen that before.

K, thnx micromass!

 

[Deveno]

the reason why the ring (Q,+,.) is described as a tuple, is to distinguish the usual ring Q from other rings that might exist on the same set with different operations.

a set is just a set. the only thing that distinguishes one set from another set (purely as a set) is its contents (its elements). but sets can have algebraic structure on them as well.

if it is less confusing to you, you can think of the ring Q as a subset of:

Q x (QxQxQ) x (QxQxQ) where + is the subset of QxQxQ: (a,b,c) ∈ + means c = a+b, and . is the subset of QxQxQ where (a,b,c) ∈ . means c = ab.

(so (3,4,7) is an element of +, but (3,4,8) is not).

it turns out that there is just one element in + of the form (a,b,_), namely, (a,b,a+b). such a subset of QxQxQ is called a function from QxQ→Q. in practical terms, name the first two coordinates, and you determine uniquely the third. perhaps it would be more like other parts of mathematics if we wrote +(a,b) instead of a+b, but that's history and tradition for ya.

 

[bentley4]

Thnx Devano!