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)

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

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)

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)

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?

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

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)

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)

subtraction?

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 76.23.246.56 (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 71.62.156.220 (talk) 19:33, 6 August 2014 (UTC)

Solutions to unital ambiguation

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)

Should this article be deleted?

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)

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)

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)

Proposed revision

This is a proposed revision: User:TMM53/Subrings-2023-03-21

I will revise the article in the near future.

This summary describes revisions

• Corrected errors in terminology. Wikipedial guideline states correct term "ring" should replace incorrect term "ring with identity." Wikipedia:Manual of Style/Mathematics
• Removed incorrect content. Subrng content was incorrectly placed in the subring article which conflicts with Wikipedia terminology guidelines Wikipedia:Manual of Style/Mathematics. This content could be placed in the correct article rng (algebra).
• Added required inline citations and needed content to introduce the required inline citations.Wikipedia:Inline citation
• Removed "Profile by commutative subrings" because the best evidence indicates that there will never be a verifiable source for this content and Wikipedia recommends the removal of this content Wikipedia:Verifiability. The reasons are 1) I challenge the accuracy of this content, 2) it lacks any citations, 3) the writing is unclear e.g. "profiled by a variety," 4) I have completed a comprehensive multiple book and online searches and cannot find a source to verify this content, 5) I examined the listed available references in the article, and I find no source 6) providing a mathematical proof to substitute for the lack of a source is not permitted Wikipedia:No original research and 7) no one has responded to a request for inline citations since 2018.
• Edited section on ideals. The revised content introducing the concepts of homomorphism, ideal and subring closely follows the approach used in major textbooks cited in the revision. Removal of prior content related to ideals as proper subrings was also required due to Wikipedia terminology guidelines.Wikipedia:Manual of Style/Mathematics.
• Corrected errors and missing content. Examples are "subring of R is a subset" (a semantic error), "preserves the structure" (structure not defined), sentences written in parentheses (writing style), etc.
• Introduced simplified notation in the notation section.
• Introduced additional content.

This content was removed due to terminology error.

Wikipedia:Manual of Style/Mathematics

" For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself."

"If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

• The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z have no multiplicative identity."

This content was removed due to no source being available.

Wikipedia:Inline citation Wikipedia:Verifiability Wikipedia:No original research

"A ring may be profiled[clarification needed] by the variety of commutative subrings that it hosts:

The quaternion ring H contains only the complex plane as a planar subring The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, as well as the ordinary complex plane The ring of 3 × 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 × 3 matrices." TMM53 (talk) 07:04, 21 March 2023 (UTC)

First, the new version definitely looks better so that should be a right direction. As for unsourced content, since the article isn’t about a real person and there is no chance of hurting anyone by containing possibly incorrect content, it’s probably better to use the citation needed template instead of just removing content (unless we are sure the content is wrong).

Also, there should be a section on a pure subring, I think. It’s an important notion and deserves to be mentioned. —- Taku (talk) 08:37, 21 March 2023 (UTC)

This is in pure submodule that “pure subring” must be defined. D.Lazard (talk) 12:49, 21 March 2023 (UTC)

I wouldn’t say "this is in ...". Even though pure subring redirects there, the article doesn’t really have a discussion of that. Also, the discussion of a special case can be important. For example, a faithfully flat ring homomorphism is just a special case of a faithfully flat module but that special case is important to be discussed independently. A pure subring (also called pure extension), in my opinion, is such a case.

An important general principle is that a subring tends not to inherit properties of an ambient ring; e.g., a subring of a Noetherian ring is usually not Noetherian. (By the way, this sort of things should be mentioned in the article). This deficiency? does however not occur for pure subrings, again the important point to make. —- Taku (talk) 19:12, 21 March 2023 (UTC)

I have just even found a paper exactly on the topic [1]. —- Taku (talk) 19:15, 21 March 2023 (UTC)

I agree that "pure subrings" is a notable topic, that must be described in Wikipedia. But this is an advanced and specialized topic that would be misplaced in this general and rather elementary article. This is another question of whether pure subrings and pure submodules should be in the same article or not. For the moment, pure subrings are not defined. So the first task is to describe them in Pure submodule. When this will be done, we will can discuss whether a split must be done and whether a mention of pure subrings must be added here. For the moment, there is not sufficient information in Wikipedia for allowing non-specialists to have an opinion. D.Lazard (talk) 21:40, 21 March 2023 (UTC)

Thanks to everyone. The updated revision is User:TMM53/Subrings-2023-03-21. I added a non-Netherian subring of a Noetherian ring is possible, retained and explained by example the quaternion rings, coquaternion rings, the nilpotent elements, and related subrings, and leave the issue of pure subring/pure submodule to future editors. TMM53 (talk) 04:58, 22 March 2023 (UTC)

To D.Lazard: it seems a question of how much results on ring extensions (integral extensions, faithfully flat ones, ...) be on this article. If we were to discuss going up and going down and such, then surely pure extensions are also not tout of place. No? —- Taku (talk) 10:42, 23 March 2023 (UTC)

Before discussing which classes of examples belong to this article, one must have an article that describes correctly the generalities. For the moment both the article and TMM53’draft do not satisfy this requirement.

The main problem with the article is that it has been written before the consensus that, in Wikipedia, a ring has a multiplicative identity, and, otherwise, it is a rng. So all the stuff about rngs must be moved into a specific section possibly called “Subrng”, where the differences with the case of rings is clearly explained. For example, if R and S are two rings, the natural inclusion of R into $R\times S$ makes R a subrng of $R\times S,$ but not a subring.

IMO the draft is worse than the current article. It contains many nonsensical sentences. In particular, the whole section “Notations” defines very vaguely unusual concepts. I am unable to understand what is meant by any of the sentence of this section. Nevertheless the phrase "ring R has set R” lets suppose that the used definition of a ring is not that of the article Ring, where a ring is a set with an additional structure. Also, the formula $\mathbb {C} \subset \mathbb {H} \subset \mathbb {M} _{2}(\mathbb {C} )$ is misleading and nonsensical, as $\mathbb {H}$ contains infinitely many subrings isomorphic to $\mathbb {C} ,$ none being naturally identified to $\mathbb {C} .$ The same is true for the other inclusion.

The section “Lying over theorem” does not belong to this article, since its proof needs a technology that is not in this article. Also, this section refers implicitely to Integral extension, although this article is to linked to in the section.

For transforming the dratf into an acceptable stub, almost every sentence must be changed. So, it seems much easier to improve step by step the present article than to replace it by something that is not better. D.Lazard (talk) 18:05, 23 March 2023 (UTC)

These are 3 steps of a step-by-step method to correct this article's problems. I will check back in about 4 weeks (27 April 2023). If more time is needed, let me know. Thanks.

====Step 1: Add inline sources====

Wikipedia:Inline citation

The ring $\mathbb {Z}$ and its quotients $\mathbb {Z} /n\mathbb {Z}$ have no subrings (with multiplicative identity) other than the full ring.

Every ring has a unique smallest subring, isomorphic to some ring $\mathbb {Z} /n\mathbb {Z}$ with n a nonnegative integer (see characteristic). The integers $\mathbb {Z}$ correspond to n = 0 in this statement, since $\mathbb {Z}$ is isomorphic to $\mathbb {Z} /0\mathbb {Z}$.

The subring test is a theorem that states that for any ring R, a subset S of R is a subring if and only if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.

If I is a prime ideal of a commutative ring R, then the intersection of I with any subring S of R remains prime in S. In this case one says that I lies over I ∩ S. The situation is more complicated when R is not commutative.

====Step 2: Remove rng content====

Wikipedia:Manual of Style/Mathematics#Algebra

Wikipedia guideline states that an article should not attempt to cover both ring and rng content. See footnote #1: "Currently, ring (mathematics) and related articles attempt to cover both unital rings and non-unital rings: they do not consistently implement this interpretation. This attempt to cover multiple meanings violates WP:DICT#Major differences (homographs)." This content would be removed:

For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

Proper ideals are subrings (without unity) that are closed under both left and right multiplication by elements of R.

If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.

The proper ideals of Z have no multiplicative identity.

====Step 3: Remove content====

This content may not be appropriate for this article. Proposed revision is to remove this from this article.

If I is a prime ideal of a commutative ring R, then the intersection of I with any subring S of R remains prime in S. In this case one says that I lies over I ∩ S. The situation is more complicated when R is not commutative.

==Notes==

==References==

• Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3.
• Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. ISBN 0-471-43334-9.
• Hartley, B.; Hawkes, T.O. (1974). Rings, modules and linear algebra: a further course in algebra describing the structure of Abelian groups and canonical forms of matrices through the study of rings and modules. London: Chapman & Hall. ISBN 978-0412098109.
• Jacobson, Nathan (1989). Basic algebra (2nd ed.). New York: W.H. Freeman. ISBN 0-7167-1480-9.
• Kuz'min, Leonid Viktorovich (2002). Encyclopaedia of mathematics. Berlin: Springer-Verlag. ISBN 1402006098.
• Lang, Serge (2002). Algebra (3 ed.). New York. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)}
• Larsen, Max D.; McCarthy, Paul J. (1971). Multiplicative theory of ideals (PDF). New York: Academic Press. ISBN 978-0124368507.
• Rosenfeld, Boris (1997). Geometry of Lie Groups. Boston, MA: Springer US. ISBN 978-1-4419-4769-7.
• David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6.

TMM53 (talk) 06:49, 30 March 2023 (UTC)

By “step by step”, I meant making edits one after the other. For example, for adding citations, doing that in different edits. As the citations that you propose are clearly convenient, be free to add them without waiting. Similarly for removing the sentence about prime ideals, since they are not defined in this article.

About the content on rng, I agree that a part must be removed, but another part must be kept as caveat for avoiding confusion. The best solution would probally be to create a section “Sub-rng” where all stuff about rngs could be moved and rewritten. D.Lazard (talk) 08:22, 30 March 2023 (UTC)

The next proposal as advised is to create a section Sub-rng. All content related to sub-rng is in this section. I added Wikipedia's definition of rng otherwise the reader will be confused by this section. This is how the revised section appears:

Sub-rng

A structure satisfying all the ring axioms except the existence of a multiplicative identity is called a rng. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With the definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

Proper ideals are subrings (without unity) that are closed under both left and right multiplication by elements of R.

If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

• The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z have no multiplicative identity.

TMM53 (talk) 19:19, 19 April 2023 (UTC)

Subring test must have extra condition of it containing the 1

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)

Sorry, I exchanged the old and the new versions in my mind. D.Lazard (talk) 14:19, 26 January 2024 (UTC)

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)

Subring test error

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)

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)