Talk:Subring
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I agree that conventions being followed are good, but it would seem to me that stating the most general definition first (if you can define what general means, and in this case we can), and then clarifying other possible definitions would be best. —The preceding unsigned comment was added by Jondice (talk • contribs) 22:27, 4 December 2006 (UTC)
- Stating the most general definition first is not always appropriate. Doing so here would lead to a definition that is inconsistent with our ring conventions. I think the most important thing is that our definition be consistent with the one at ring. Alternative definitions and conventions can and should be listed, as they are now. -- Fropuff 03:06, 5 December 2006 (UTC)
[edit] Two inequivalent definitions
I think the subring article states two inequivalent definitions. (Recall that a ring is always assumed to have identity, in wikipedia.)
Definition 1 in the first paragraph of the subring article: "a subring is a subset of a ring, which is itself a ring under the same binary operations."
Definition 2 in the second paragraph: "we say that a subset S of R is a subring of R if it is a ring under the restriction of + and * to S, and contains the same unity as R"
For example, consider the ring R of all integer pairs and the subset S of R consisting of all integer pairs with second coordinate being zero. R and S are (unital) rings because R has identity (1,1) and S has identity (1,0). S is a subring of R according to definition 1, but it is not a subring according to definition 2.
But the subring article seems to claim that the two definitions are equivalent.
-- Novwik, October 30th 2005
I'm not sure this is a correct example, by the following argument: R = Z x Z, while S = Z x 0, 0 the nullring with just one element: 0. In the nullring, 0 = 1, so in fact S is not a subset of R: If it was, then S should also contain for instance (4,1), being equal to (4,0), but it doesn't. At least, if the example is correct, it may be pathological. (When thinking about it, my argument seem to be humbug; I leave it as an exercise to delete it.)
-- Somebody
[edit] The ring Z does has subrings
the article states: The ring Z has no subrings other than itself.
This is not a true statement. Consider the set of even integers; this is a subset of Z and is closed under addition and multiplication, and both operations are associative and commutative in this subset. The distributive law also holds. It also has the zero element of Z, as well as additive inverses. Thus by defintion, the even integers form a subring of Z.
As another example, the trivial ring {0_R} is a subring of any ring; the integers being no exception.
—The preceding unsigned comment was added by Thearn4 (talk • contribs) 20:35, 2 March 2007 (UTC).
- No, it doesn't. Try reading the article for an explanation. -- Fropuff 21:13, 2 March 2007 (UTC)
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- This is actually a matter of convention: for some authors, rings and algebras are not necessarily unital - see ring. This should probably be mentioned here as well. Geometry guy 22:11, 2 March 2007 (UTC)
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- It is mentioned in the second sentence. For some reason, however, this article stills seems to confuse people. I'm not sure how to make it less confusing. -- Fropuff 00:58, 3 March 2007 (UTC)
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[edit] When do commutative subrings cover a ring?
Always. Every element x of a ring lies in a subring generated by {x} which includes the powers of x. Since under even the weakest of associativity premises these powers of x commute with one another, this subring is commutative. Clearly the ring is covered by such subrings. This statement corrects my previous comment.Rgdboer (talk) 02:42, 22 February 2008 (UTC)
