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I have read that subgroups, subrings, submodules, etc. are substructures.

But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals and Noetherian modules are defined with submodules. Isn't it awkward? Why does submodule correspond to ideal, not subring? Is there any definition of Noetherian with subrings?

As I'm studying commutative algebra, it looks like ideals are more important than subrings. But why is it ideal, not subring (which seems to correspond to all other substructures)? Though I am not very familiar with pseudo-rings, is it true that ideal is a sub-pseudo-ring (or sub-rng) and thus we can view ideal as a kind of substructure?

user26857
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Gobi
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6 Answers6

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The "right" notion of a substructure of an algebraic gadget is the kernel of a homomorphism. For abelian groups, and more generally modules, these are subgroups, respectively submodules. For groups, we need normal subgroups. For rings, we need ideals.

Sammy Black
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    Though normal subgroups of groups and ideals of rings are very important, and though I agree that it is good to view them as kernels of morphisms out of the object, this does not seem to be what is described by the notion of *[substructure](http://en.wikipedia.org/wiki/Substructure)*. $$ $$ By the way, there is a word doubling on your website ("are are"). – Rasmus May 30 '11 at 09:09
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    This "answer" is entirely wrong. For starters, "most" homomorphisms between algebraic gadgets *have* no notion of kernel. – goblin GONE Feb 17 '15 at 05:32
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    Context! A lot of important categories are abelian, especially ones that the OP is/has been looking at. – Sammy Black Feb 17 '15 at 18:53
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Any ring $R$ is a module over itself, in the obvious manner: $R$ is an abelian group, and we define $a\cdot b$ for $a\in R$ and $b\in R$ to be $ab$.

A submodule of $R$-as-an-$R$-module is precisely an ideal of $R$ (work out the relevant definitions to see this). Thus, the definition of a Noetherian ring is really saying that it is a Noetherian module over itself. Indeed, one of the important points about commutative algebra that I learned from Atiyah-Macdonald is that given a ring $R$, we are interested in both ideals $I\subset R$ and quotient rings $R/I$ of $R$, and introducing the concept of an $R$-module - which $I$, $R$, and $R/I$ are all examples of - allow us to treat everything on a roughly equal footing.

I don't believe that rings satisfying the ascending chain condition for subrings, which we might call "subring-Noetherian", are studied or have nice properties, but I could be wrong about this.

I think it's fair to say that ideals are more important than subrings, but subrings are still integral (ha ha) for commutative algebra. You are correct that if we don't require our rings to have a multiplicative identity, ideals are subrings, i.e. "sub-rngs".

Zev Chonoles
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Both ideals and subrings are important in algebra. But generally there is no close relationship between congruences on $\rm\,A\,$ and subalgebras of $\rm\,A.\,$ Instead, generally congruences are related to subalgebras of the square $\rm\,A^2,\,$ e.g. see here. Rings (and groups) are special in that their congruences are determined by a single congruence class - which has the effect of collapsing the relationship of congruences with subalgebras from $\rm\,A^2\,$ down to $\rm\,A.\,$

One way to better understand the importance of ideals is to study other algebras whose congruences are determined by a single congruence class - so-called ideal determined varieties. They are characterized by two properties of congruences, being $\,0$-regular and permutable, e.g. see below for an entry point into literature on such topics.

ON SUBTRACTIVE VARIETIES IV: DEFINABILITY OF PRINCIPAL IDEALS

Paolo Agliano and Aldo Ursini

$0.\ \ $ Foreword

We have been asked the following questions:

(a) $\ $ What are ideals in universal algebra good for?
(b) $\ $ What are subtractive varieties good for?
(c) $\ $ Is there a reason to study definability of principal ideals?

Being in the middle of a project in subtractive varieties, this seems the right place to address them.

To (a). The notion of ideal in general algebra [13], [17], [22] aims at recapturing some essential properties of the congruence classes of $0$, for some given constant $0$. It encompasses: normal subgroups, ideals in rings or operator groups, filters in Boolean or Heyting algebras, ideals in Banach algebra, in l-groups and in many more classical settings. In a sense it is a luxury, if one is satisfied with the notion of "congruence class of $0$". Thus in part this question might become: Why ideals in rings? Why normal subgroups in groups? Why filters in Boolean algebras?, and many more. We do not feel like attempting any answer to those questions. In another sense, question (a) suggests similar questions: What are subalgebras in universal algebra good for? and many more. Possibly, the whole enterprise called "universal algebra" is there to answer such questions?

Having said that, it is clear that the most proper setting for a theory of ideals is that of ideal determined classes (namely, when mapping a congruence E to its $0$-class $\,0/E\,$ establishes a lattice isomorphism between the congruence lattice and the ideal lattice). The first paper in this direction [22] bore that in its title.

It comes out that -- for a variety $V$ -- being ideal determined is the conjunction of two independent features:

$1$. $V$ has $0$-regular congruences, namely for any congruences $E,E'$ of any member of $V,\,$ from $\,0/E = 0/E'$ it follows $E = E'$.

$2$. $V$ has $0$-permutable congruences, namely for any congruences $E,E'$ of any member of $V,\,$ if $\ 0\, E\, y\, E'\, x,\,$ then for some $z,\ 0\, E'\, z\, E\, x$.

Bill Dubuque
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    That looks like a nice paper, but really, the only answer to a question which asks "What is $X$ good for?" is "Absolutely nothing! *uh* Say it again, yeaahhhh!". – Gunnar Þór Magnússon May 30 '11 at 19:44
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This answer is (by intent) very similar to Zev Chonoles' answer, but I have tried to put it in terminology a bit more similar to that of the question.

When one wants to study a ring $R$, there are (at least) two avenues of approach. The more naive approach is to study $R$ as a ring structure. In this sense, you are correct: a substructure of $R$ is precisely a subring of $R$. However, while the study of ring structures is undoubtedly important, it turns out that it is even more useful to study structures of $R$-modules. [One way to obtain a language of $R$-modules is to take the language of abelian groups and add a 1-ary function symbol for every element of $R$.] When one considers $R$ as having the structure of an $R$-module rather than a ring, then its substructures are precisely its ideals.

Why is it more useful to study $R$-modules than rings? It's hard to give a precise answer to that question (especially since the study of ring structures, subrings, etc. is in fact very useful and important), but one way of thinking about this is that when you are looking at $R$-modules, there is a nice notion of "quotient." On the other hand, if you have $S$ as a subring of $R$, then in particular $1 \in S$; so if you try to take a "quotient ring" $R/S$, you have to set $1=0$ in the quotient ring, i.e., you get the zero ring.

Charles Staats
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So I'm a bit late to the party on this one, but let me add something.

Proposition. Let $R$ denote a ring. Then $R$ probably has "too many" subrings.

Okay - so its not quite a genuine proposition. But dammit, its still true! For instance, if $R$ is a ring, then there's always a unique homomorphism $\mathbb{Z} \rightarrow R,$ and the image of this homomorphism is always a subring of $R$ (which is related to the characteristic of $R$). So if you're looking at the subrings of $\mathbb{C}$, for example, you wind up having to think about $\mathbb{Z}$, $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{2}]$, etc. There's just too many of them! The trick is to ignore most of those subrings by dealing with subalgebras instead. For instance, viewing $\mathbb{C}$ as an $\mathbb{R}$-algebra, we see that it has has precisely two subalgebras: namely $\mathbb{R}$, and $\mathbb{C}$ itself.

Now on the other hand, if you're dealing with ideals, there's no need to move to algebras; the ideals of a ring $R$ don't change if we suddenly start viewing $R$ as a something-algebra. For instance, the ring $\mathbb{C}$ has precisely two ideals - namely $\{0\}$ and $\mathbb{C}$ - and this doesn't change if we start viewing it as an $\mathbb{R}$-algebra or a $\mathbb{C}$-algebra.

In motto form: ideals are more important than subrings because they're stable under "change of viewpoint."

goblin GONE
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To start, it's worth being a bit clearer on where ideals come from. To quotient a set, we need an equivalence relation on that set. If that set is equipped with algebraic operations, e.g. it is a ring, then for the result to be the same sort of algebraic structure, the equivalence relation needs to respect the algebraic operations. An equivalence relation that respects the relevant algebraic operations is called a congruence. Given some congruence, $\approx$, on a ring, $R$, we can form the quotient ring $R/{\approx}$ simply by quotienting the underlying set. The original operations will be well-defined because $\approx$ is a congruence (and not just an equivalence relation).

Because $\approx$ is a congruence, we have that $r\approx s\iff r-s\approx 0$. The equivalence class of $0$, i.e. $[0]=\{r\in R\mid r\approx 0\}$ is exactly an ideal. That is, every congruence gives rise to an ideal by considering the equivalence class of $0$, and every ideal, $I$, gives rise to a congruence via $r\sim s\iff r-s\in I$. This is related to the first paragraph of Bill Dubuque's answer. This is also exactly the same story for normal subgroups. For modules, it just so happens1 that the analogue of an ideal is itself a submodule. So it's not that for modules we care about substructures and for rings we care about ideals. We care about the same thing in both cases, it just so happens that the "ideals" of modules coincide with submodules.

Before I continue, one interesting fact is that subobjects and quotient objects are actually dual (in the categorical sense) notions (at least in regular categories which includes any category of algebraic structures, such as the category of rings).

As indicated in Zev Chonoles answer, subobjects are far from unimportant. Nevertheless, ideals take a lot more of the spotlight. I'll give two reasons why that may be the case. I'll start with a more sociological answer, and then I'll give a more technical answer.

The sociological answer is simply that ideals are a more challenging to understand concept than subrings, so more time is spent on them. For any algebraic object (for a technical sense of algebraic, but one that captures most examples with the notable exception of fields), a subobject is exactly the thing you think it would be. Namely, a subset for which the original operations are closed. Quotient sets in general are harder to understand than subsets, and ideals are often presented in a way that pulls the definition out of a hat. Even when the connections to congruences are made clear, it is still fairly complicated. The notion is also not as uniform across algebraic structures. While quotienting by a congruence is something that is meaningful for any algebraic object, there is not always a distinguished constant to consider the equivalence class of like $0$, and even if there is it isn't always helpful. In the examples above, all relied on the group structure of the algebraic structures.

Early examples are also misleading. The "ideals" of vector spaces (and more generally modules) are exactly sub-vector-spaces. For groups, they are normal subgroups. These examples make it look like we should be quotienting by substructures. There's also an aesthetic niceness of being able to "divide" vector spaces. The ring case looks out of place since an ideal isn't a subring, not even a special type of subring, and thus we can't "divide" rings. This likely leads some to think that a ring with identity is the wrong notion, and we should be considering rings without identity, often called rngs, because ideals are subrngs.

On the more technical side, every algebraic structure is a quotient of a free structure. This is almost the defining property of being "algebraic". This is, of course, the basis of the notion of a presentation. From this perspective, we care more about ideals because they to congruences and thus quotients, and quotients are more fundamental to algebraic structure than subobjects.

Another fact is that while categorical limits (which include subobjects and products) are given in a simple way for any algebraic category, colimits, which will typically involve quotients and which quotients are an example, need not be. For example, the (categorical) product of two rings is just defined by the (categorical, i.e. cartesian) product of their underlying sets with the operations defined component-wise. This works for any algebraic structure, e.g. the (direct) product of groups or products of lattices. The categorical coproduct, on the other hand, while it always exists for algebraic categories, is not so simple. The coproduct of two groups or two rings is not at all the coproduct (i.e. disjoint union) of their underlying sets. Indeed, the easiest approach is to consider free objects, where the answer is easy, and then quotient that, hence the "free product" of groups which is the coproduct. The upshot of this is that a lot of the time colimits are going to be where interesting things happen and where different algebraic structures really show different behavior. For example, for abelian groups and modules finite products and finite coproducts coincide, but this is certainly not the case for (arbitrary) groups and rings.

To be clear, a lot of ring theory is secretly module theory. Similarly, a lot of group theory is secretly the theory of $G$-sets. Groups are to rings as $G$-sets are to modules. Many results are easier and more intuitive when you view them as results about modules which can then be specialized to rings. Categories of modules are also abelian categories, unlike the category of rings, which provides a lot of nice laws and structure which simplifies things immensely.

1 This is not completely an accident and follows from the fact that categories of modules are abelian.

Derek Elkins left SE
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