Talk:Subring

message: 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)
thread_title:  Untitled 

message: 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 107.15.35.205 (talk) 10:42, 8 May 2015 (UTC)
thread_title:  Two inequivalent definitions 

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

message: 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)
thread_title: When do commutative subrings cover a ring?

message: 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 (talk • contribs) 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)
replies: text:  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)
thread_title: Spurious 'ref' request removed

message: This article has several problems: 
: 1) Wikipedia:Verifiability: There are no inline citations despite the request made in 2018, the huge burden for an editor to search and find sources for all content is unreasonable, and most content appears original, not from a source.  The best evidence is that this problem will never be corrected.
: 2) Wikipedia:Manual of Style/Mathematics: This article fails to comply with the Wikipedia guideline that ring has an identity element and rng does not.  The failure to comply with this guideline leads to a confusing article.
: 3) This article is redundant.  The article Ring (mathematics) contains a section subring.  For content related to rng there is an article Rng (algebra).
: 4) If I were to edit this article, remove original non-sourced content and retain only content that I can verify with a source, the article would be very brief but meet Wikipedia guidelines.
These are some options
: 1) Request article deletion. Wikipedia:Articles for deletion 
: 2) The original author can provide inline citation of sources.  Not likely as no response since 2018.
: 3) I can edit the page and generate a brief article.
What do other editors think? TMM53 (talk) 05:16, 17 March 2023 (UTC)
replies: text: The article lists some references. So I don’t think the sourcing is much of issue. As for inline citations, yes, the article should include more. But statements or facts stated here are rather trivial; so they are verifiable (by logic) and so we don’t need to delete them. There are some stuffs that would be too distracting to include in the ring article so I think it makes sense that this article exists. —- Taku (talk) 06:48, 17 March 2023 (UTC)
replies: text: Hello.  I appreciate your response.  I will follow your recommendation to not recommend the deletion of this article.  I agree that the general references are good.  The typical reader for this article is a person who has likely never completed a course in abstract algebra.  According to Wikipedia guidelines, we as editors should identify "any material challenged or likely to be challenged" by this typical reader and ensure that an inline source accompanies this content.  In my view, there is content that meets this criteria, so inline references are needed.  The other requirement is that a reference "provides complete, formatted detail about the source, so that anyone reading the article can find it and verify it."  This means citing specific pages of a book, not citing the book alone (Wikipedia:Citing sources).  I will propose an edit.  Thanks again for your response. TMM53 (talk) 01:14, 18 March 2023 (UTC)
thread_title:  Should this article be deleted? 

message: Dummit and Foot mentions that the test doesn't assume the ring has a 1. You can see this discussed further here. If the ring has a 1, then it must be the same 1 as the parent ring as defined in this article. Okoyos (talk) 02:56, 26 January 2024 (UTC)
replies: text: Sorry, I exchanged the old and the new versions in my mind. D.Lazard (talk) 14:19, 26 January 2024 (UTC)

text: Counter example, take Z_6 with the subring {0,2,4}, 2+2=4, 2+4=0, and 4+4=2, thus it's closed, also 4*2=2, and 4*4=4, thus any non-zero element in the subring multiplied by 4 is itself, thus 4 is the identity element. The subring doesn't need its parent rings identity to have its own. 70.54.110.23 (talk) 12:21, 31 January 2024 (UTC)
thread_title:  Subring test must have extra condition of it containing the 1 

message: A subring does not need the multiplicative identity of R in it, in fact it can have a multiplicative identity that isn't the multiplicative identity of R. Take Z_6 with the subring {0,2,4}, 2+2=4, 2+4=0, and 4+4=2, thus it's closed, also 4*2=2, and 4*4=4, thus any non-zero element in the subring multiplied by 4 is itself, thus 4 is the identity element. 
I believe it should read the additive identity must be in S for S to be a subring of R. 70.54.110.23 (talk) 12:20, 31 January 2024 (UTC)
replies: text: In your example, {0,2,4} is not a subring, and does not pass the subring test, as it does not contain the multiplicative identity of Z_6 (which is 1). There is no error in the article, if you do not change the given definition of a subring. In summary, {0,2,4} is not a subring of Z_6, but it is a ring that is a subrng of Z_6. D.Lazard (talk) 14:50, 31 January 2024 (UTC)
thread_title:  Subring test error