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WikiProject Mathematics (Rated Start-class, Mid-importance)
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 Field: Algebra

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 (talkcontribs) 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)

Two inequivalent definitions[edit]

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

These are definitely inequivalent definitions. Consider, as another example, Z/6Z. Then 2Z/6Z would be a subring with respect to Definition 2, but only because it would use a different multiplicative identity. In 2Z/6Z, for example, 4+6Z is a multiplicative identity: (4+6Z)(0+6Z) = 0+6Z, (4+6Z)(2+6Z) = 2+6Z, and (4+6Z)(4+6Z) = 4+6Z. It would not be a subring with respect to Definition 1, however, since it doesn't contain the multiplicative identity 1+6Z from the ambient ring Z/6Z.

Separately, taking R := Z×Z and S := Z×{0} seems reasonable. In S, we need not think of {0} as a ring in which 0=1. Here, {0} is the subset of Z containing only the integer zero. All we need to observe is that S is indeed a subset of R, and it's one closed under the addition, subtraction, and multiplication operations of R. If you view S as a product of Z with the trivial ring, then it is indeed not even a subset of R, but I don't read that as the intended interpretation of S.

-- Somebody Else — Preceding unsigned comment added by (talk) 10:42, 8 May 2015 (UTC)

The ring Z does has subrings[edit]

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 (talkcontribs) 20:35, 2 March 2007 (UTC).

No, it doesn't. Try reading the article for an explanation. -- Fropuff 21:13, 2 March 2007 (UTC)
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)
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)
Because rings are unital, Z has no subrings other than itself. ᛭ LokiClock (talk) 04:43, 7 October 2011 (UTC)

When do commutative subrings cover a ring?[edit]

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)

Spurious 'ref' request removed[edit]

The statement:

Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity (if it exists).

was tagged with 'fact' -- this is absurd; if a given author doesn't require rings to have a multiplicative identity in the first place, how could they possibly then require a subring to possess one?—Preceding unsigned comment added by Zero sharp (talkcontribs) 19:52, 22 September 2008

Agreed. But this ring-with-a-one thing is a perennial cause of confusion. For example, {(x,0)} is a subrng of ZxZ, and it has a one, but it isn't a subr1ng, as (1,0) /= (1,1). I wonder if this was in Catherine Yronwode's mind Richard Pinch (talk) 06:17, 26 September 2008 (UTC)
Fair enough, I _seriously_ doubt that was the source of the confusion here. I think as long as it's made clear that some authors require a ring to have a '1' and some don't, and if they don't then subring's don't, it's fine. BTW I like the notation 'r1ng'. Zero sharp (talk) 15:10, 29 September 2008 (UTC)


is there a particular reason the article specifies closed under subtraction as opposed to addition? in fact, the very beginning starts with stating subtraction. given that a subring is still a ring, the group nature of the additive operation makes mentioning a subtraction operation completely superfluous. i'll give a few days for someone to respond with something reasonable before i modify it. —Preceding unsigned comment added by (talk) 09:30, 4 March 2011 (UTC)

This is a very old comment, but in case other people come around and wonder the same thing, it's worth answering: closure under subtraction is a sneaky way of guaranteeing that the subring contain the additive inverses of elements. (For example, if a is in a subring S, then 0 – a is in S as well.)
If subtraction were replaced with addition in the theorem, then the non-negative integers would satisfy the hypotheses, since the set is nonempty, and also closed under both addition and multiplication. Yet the non-negative integers are not a subring of the integers. --Heath (talk) 19:33, 6 August 2014 (UTC)

Solutions to unital ambiguation[edit]

There are some problems from requiring rings & subrings to be unital. I have two solutions, one to include a section on the not-necessarily-unital generalization and its differences, and the other to include this {{hatnote}}: This article defines rings and subrings as unital. Not all statements are true for definitions that don't require a multiplicative identity. ᛭ LokiClock (talk) 04:58, 7 October 2011 (UTC)

The problem arises from the article trying to be a dictionary (and there can be no doubt that this is a problem). A Wikipedia article should cover a single topic/concept, not conflicting uses of a term (homographs). There are two clearly different concepts here, each notable and of utility in its own right:
  1. A restriction on the set R that preserves the unital ring structure (R, +, ×, 0, 1)
  2. A restriction on the set R that preserves the pseudoring structure (R, +, ×, 0)
Wikipedia has a guideline as to which to interpretation to apply the term ring, and hence subring: the first of the concepts listed here. The detailed presentation of the second case should be left to Pseudo-ring and related articles. This article should only mention that the term "subring" has a different meaning according to the author's choice of terminology. —Quondum 15:50, 16 November 2013 (UTC)