Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant description in terms of the associated (concrete) category of modules?
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$\begingroup$ I guess you are looking for descriptions of the form "every module is ...". But do you also consider descriptions of the form "every module has the property that if it has A then is has also B" appropriate (which may be considered a special case of the above)? $\endgroup$– Fred RohrerCommented Mar 25, 2014 at 11:40
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7 Answers
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A commutative noetherian local ring $R$ is regular if and only if every $R$-module has a finite projective resolution (Serre theorem).
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A ring $R$ is von Neumann regular if and only if every left (or right) $R$-module is flat.
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From http://en.wikipedia.org/wiki/Injective_module
Every submodule of every injective module is injective if and only if the ring is Artinian semisimple (Golan & Head 1991, p. 152); every factor module of every injective module is injective if and only if the ring is hereditary, (Lam 1999, Th. 3.22).
Every infinite direct sum of injective right modules is injective if and only if the ring is right Noetherian, (Lam 1999, Th 3.46). Every infinite direct product of flat right modules is flat iff the ring is left coherent, and every infinite direct product of projective right modules is projective iff the ring is right perfect and left coherent. (The latter two theorems appear in this paper by Stephen Chase. )
Another characterization of (one sided, as above) noetherianity with injective modules: every injective is direct sum of indecomposables; there is only a set (not proper class) of isomorphism types of injective indecomposables.
Dually one has many characterization results concerning projectivity; for example for a commutative integral domain, one of the many equivalent conditions to be Dedekind (Pr\"ufer) is hereditary (semi-heredirary).
QF rings: every projective is injective (on one side) iff every injective is projective (on one side) iff the ring is two-sided artinian and the finitely generated right and left modules are in natural duality (with the usual dual, hom to the ring of scalars) iff the ring is (one sided) noetherian and in the Galois correspondence "annihilator" between right and left ideals, each element is closed.
Principal ideal artinian rings: every homomorphic image is QF iff finite direct products of matrix rings over CPU rings (rings $R$ with a nilpotent element $p$ such that $pR=Rp$ and the $p^nR=Rp^n$ are all one sided ideals) iff each finitely generated module has a lattice of submodules which is finite direct product of primary lattices. [Generalizing primary decomposable to semiprimary characterizes artinian serial rings]
von Neumann regular: $\forall a\exists x:axa=a$ iff each in each finitely presented (or even only finitely generated projective) module the semilattice of finitely generated submodules is a complemented sublattice of the lattice of all submodules. Von Neumann regularity is also equivalent to all right modules being flat, and is also equivalent to all right modules being principally injective (a.k.a. "divisible" in Lam's *Lectures on modules and rings.")
Classically semisimple: every module has complemented lattice of submodules.
One can multiply examples, ad enjoy finding them in the books by Lam, Rowen, Faith, Stenstr\"om, ...
However: all the magic disappares when one notes that every Morita invariant property can be defined by a purely categorical property of the abelian category of all right modules, and when in such a category one fixes a progenerator then any property of the ring can be expressed in this language (compare semisimple rings, a Morita invariant property, versus skew fields, characterized by freenes of modules instead of injectivity / projectivity). Besides, all these characterizations are also possible in lattice theoretic terms (Hutchinson - Isbell lattice associated to a abelian category); when the property to be expressed is not Morita - invariant one needs to fix a basis in the lattice.
So the problem is not the possibility to express a ring property in the categorical language of modules (bicartesian language of abelian categories, or the language of lattice theory). The problem is to formalize the requested "elegance" of the characterization, a problem with no clear mathematical answer. Not a true formalization, but a guide might be: seach for equivalence between properties valid for all modules (or somewhat "unbounded" classes of modules: all finitely generated modules; all projetive modules; ...) and properties that can be expressed "internally" in the ring (or for example in its $2\times 2$ matrix ring, or that in any case depend on a class of modules that is "bounded", like $n$-generated modules for a fixed $n$. Example: a ring is unit regular iff the ring is vNr and perspectivity is transitive in the lattice of the $2\times 2$ matrix ring iff the ring is vNr and perpectivity is transitive in all the lattices of finitely generated submodules of a finitely presented module iff the ring is vNr and cancellation is valid in the additive monoid of isomorphism types of finitely presented modules)
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For a ring $R$, the following statements are equivalent:
(i) $R$ is semisimple;
(ii) Every left (or right) $R$-module is semisimple;
(iii) Every left (or right) $R$-module is injective;
(iv) Every left (or right) $R$-module is projective.
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A special case of what you are asking for are properties that are Morita invariant. I just copy the list from the Wikipedia article:
simple
semisimple -- [see Fred Rohrer's answer]
von Neumann regular -- [see Fred Rohrer's answer]
right (or left) Noetherian -- [see user46855's answer]
right (or left) Artinian
right (or left) self-injective
quasi-Frobenius -- [see user46855's answer]
prime
right (or left) primitive
semiprime
semiprimitive
right (or left) (semi-)hereditary -- [see user46855's answer]
right (or left) nonsingular
right (or left) coherent -- [see user46855's answer]
semiprimary
right (or left) perfect
semiperfect
semilocal
There are more in Lam's book Lectures on Modules and Rings (§18 in particular), e.g. "finite uniform dimension" or even "finite cardinality". Of course it is not said that the description in terms of the module category is always as beautiful as in some of the specific answers; sometimes it is a nice exercise to search for a "concrete" and "nice" description in terms of modules.
I take the opportunity to advertise my question MO/124856 which asks whether there is such a description for being classical.
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$\begingroup$ It might be interesting to create a mini-list that says which of these have elegant descriptions. For instance the first couple have descriptions in this very thread. $\endgroup$ Commented Mar 25, 2014 at 19:08
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1$\begingroup$ I cannot comment about elegance, but two other useful books for hunting such descriptions are: math.uni-duesseldorf.de/~wisbauer/book.pdf (systematic use of abelian categories of modules to express many classical properties of rings and modules) and C. Faith, <a href="books.google.it/… and things and a fine array of twentieth century associative algebra</a> (I bet that many mathoverflow users will like the second part of this book) $\endgroup$ Commented Mar 28, 2014 at 8:33
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If $R$ is an integral domain, then every ideal factors into primes if and only if every submodule of a projective module is projective.
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$\begingroup$ You want a constraint on the ideals. For a field, the module condition is automatically true while the condition on ideals is false for (0) and (1), the only ideals. $\endgroup$– KConradCommented Jul 4, 2014 at 3:48
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$\begingroup$ @KConrad: I edited to insert the word "proper", which deals with $(1)$. It seems to me that $(0)$ is prime and therefore (trivially) a product of primes, so I'm not clear on why you say that's also problematic. $\endgroup$ Commented Jul 4, 2014 at 15:24
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1$\begingroup$ Surely the ideal $(1)$ can be considered as the empty product of primes. $\endgroup$ Commented Jul 4, 2014 at 15:46
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$\begingroup$ @StevenLandsburg: Admittedly my whole comment was pedantic, so with that in mind I would not consider factoring ideals to be something that includes the zero ideal as an object of interest, e.g., in a Dedekind domain nobody ever considers a factorization of the zero ideal, but only factorization of nonzero ideals as products of nonzero prime ideals. You'd lose the uniqueness of prime ideals factorization in a Dedekind domain if you want to bring the zero ideal into consideration. $\endgroup$– KConradCommented Jul 4, 2014 at 18:07
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$\begingroup$ @EricWofsey : In view of your comment, I'm rolling back my edit. $\endgroup$ Commented Jul 4, 2014 at 19:18
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In a ring $R$, every submodule of a free $R$ module is free iff every ideal in $R$ is $R$ free.