Proving Subrings: Understanding the Empty Subset of a Ring

[blacklily]

My teacher put heavy emphases on acknowledging that a subset is nonempty of a ring when proving something is a subring of something.
For the final exam, she wants to put a question with empty subset of a ring, and do subring proof. The empty subset would have "nice" description so you can do closed under subtraction and closed under multiplication (but it would be meanless to pick elements from a empty subset)
I find it hard to come up with such description of a empty subset of a ring. Anybody have any ideas?

 

[StatusX]

I said something before, but it was wrong, so let me try again.

The usual definition of a subring is that it is a subset of a ring that is itself a ring with the operations induced from the original ring. You can show this is equivalent to closure under addition and multiplication, the existence of an additive and (usually) a multiplicative identity, and the existence of additive inverses. In turn, you can use the shortcut that the subset is closed under subtraction and multiplication.

However, this latter definition will only be equivalent to the others if you specify the subset is non-empty (note you don't have to do this with the other defintiions, which have existence statements, and so can't be satisfied by empty sets). This is because while the 1st definition clearly implies the second, to show the converse is true, you need at least one element in the subset, say a. Then a-a=0 must be in the ring (it has an identity) and for any x, 0-x=-x must be in the ring (it has additive inverses). Also, if you want a multiplicative identity, you'll need to be explicit about this too.

 

[blacklily]

I understand what you are saying and what you said is true. But what I want is:
For example:
Show (------) is a subring of (______).
Students would proceed the subring test using: (1) closed under subtraction (2) closed under multiplication but (------) is actually a empty subset of (______). Students forget that being nonempty is a part of the "shortcut" subring proof.
Is there such description of ring and subring that it's not so obvious the subset is empty, so when students proceed to use (1) and (2), they would be wrong (as it is meanless to pick elements from a empty set).

I guess you could avoid the problem by using the formal subring test: closed under addition and mulitplication, existence of additive element and additive inverses.

Thanks for your time.

 

[StatusX]

I don't really understand what you're asking. Are you looking for a definition of a subring that doesn't require you to explicitly require the subset to be non-empty? The first two I mentioned will do. Also, you might want multiplicative identites in your ring, in which case requiring a subset has this (and, moreover, that it matches the identity for the original ring, which isn't necesarilly the case) would only be possible if the subset is non-empty. In general, exstence requirements are never true of empty sets, while universal statements always are (you can see this is compatible with the usual way of converting one to the other with negations).

 

[Hurkyl]

I think he's looking for a tricky example -- one where it's not immediately obvious that it's the empty set. I.E. the problem is:

Let R be {some ring}
Let S be {elements of R that have the property P}
Is S a subring of R?​

where S is actually the empty set, but one could prove that S is closed under subtraction and multiplication without noticing that S is empty.
I have a problem though; the following statement is not a theorem:

If S is a nonempty subset of a ring R that is closed under subtraction and multiplication, then S is a subring of R​

An example is the subset of Z[x] consisting of all multiples of x. That is closed under subtraction and multiplication, but it does not contain 1 so it's not even a ring.

 

[JasonRox]

I think he's looking for a tricky example -- one where it's not immediately obvious that it's the empty set. I.E. the problem is:

Let R be {some ring}
Let S be {elements of R that have the property P}
Is S a subring of R?​

where S is actually the empty set, but one could prove that S is closed under subtraction and multiplication without noticing that S is empty.


There is a slight problem though: being
(1) a nonempty subset
(2) closed under subtraction
(3) closed under multiplication

isn't enough to be a subring -- the set must also contain 1. (well, I hear rumors that some prefer to define rings without requiring 1, but I've never actually seen that myself. I think "rng" is the popular term for such a thing)

Wow, that'd be a nice question.

 

[blacklily]

I have a problem though; the following statement is not a theorem:

If S is a nonempty subset of a ring R that is closed under subtraction and multiplication, then S is a subring of R​

An example is the subset of Z[x] consisting of all multiples of x. That is closed under subtraction and multiplication, but it does not contain 1 so it's not even a ring.

You don't need to have multiplicative identity, ie 1, for something to be a ring.
Or am I wrong?

I think he's looking for a tricky example -- one where it's not immediately obvious that it's the empty set. I.E. the problem is:

Let R be {some ring}
Let S be {elements of R that have the property P}
Is S a subring of R?

where S is actually the empty set, but one could prove that S is closed under subtraction and multiplication without noticing that S is empty.

Edit:
Thanks for explaining my question, you made it that much easier to understand. haha...

 

[Hurkyl]

You don't need to have multiplicative identity, ie 1, for something to be a ring.
Or am I wrong?

I've heard rumors that some people define rings so that they don't have to have 1, but I've never actually seen anyone do it. I've always seen such a thing called a rng (ring without identity), or possibly an algebra.

 

[blacklily]

I see. In any case, I am use to the definition of a basic ring without the 1 identity. A ring with 1 is an additional property.
I guess your example wouldn't work in my class.
Any thoughts on the orginal question?

 

[HallsofIvy]

I've heard rumors that some people define rings so that they don't have to have 1, but I've never actually seen anyone do it. I've always seen such a thing called a rng (ring without identity), or possibly an algebra.

Oh, sometimes mathematicians are just too cute!

 

[Hurkyl]

Oh, sometimes mathematicians are just too cute!

If that's not enough for you, we also have a name for when we don't require additive inverses: a rig (ring without negatives).

 

[AlphaNumeric]

I've heard rumors that some people define rings so that they don't have to have 1, but I've never actually seen anyone do it. I've always seen such a thing called a rng (ring without identity), or possibly an algebra.

Robinson in 'An Introduction to Abstract Algebra' defines a ring without talking about the identity, but within 2/3 of a page of the chapter on rings he mentions the identity and from then on always says "Let R be a ring with identity..." before going onto more advanced stuff (domains etc) which require the identity.

If that's not enough for you, we also have a name for when we don't require additive inverses: a rig (ring without negatives).

I take it no one considers any rg structures, it'd be too hard to say during a talk