Finitely Generated Module

In the intricate tapestry of mathematics, particularly within the realm of algebra, one frequently encounters the concept of a module. Modules, in essence, generalize the familiar notion of a vector space by allowing the scalars to come from a more general algebraic structure: a ring, rather than strictly a field. Among the various classifications of modules, the "finitely generated module" stands out as a fundamental concept, indicating a module that, despite its potential complexity, can be entirely constructed or described from a manageable, finite collection of its own elements.

A module is declared "finitely generated" if it possesses a finite generating set. This isn't rocket science, merely a statement of efficiency. Over a ring R, such a module might also be referred to as a "finite R-module," "finite over R," or even a "module of finite type." It seems mathematicians enjoy having options for nomenclature, even if they all point to the same rather straightforward idea.

This foundational concept branches out into several related, though distinct, ideas that refine our understanding of module structure. These include finitely cogenerated modules, finitely presented modules, finitely related modules, and coherent modules. Each of these definitions adds a layer of structural constraint, making the module behave in progressively more "well-behaved" ways. Interestingly, over a Noetherian ring – a class of rings that exhibit a certain predictable regularity – these seemingly distinct concepts of finitely generated, finitely presented, and coherent modules remarkably converge, becoming one and the same. It's almost as if the universe occasionally decides to simplify things for us.

To put this in a more digestible context, consider some familiar algebraic structures. A finitely generated module over a field is nothing more, nothing less, than a finite-dimensional vector space. This is the simplest case, where the "scalars" behave perfectly. Similarly, a finitely generated module over the integers (which form a ring, not a field) is precisely what is known as a finitely generated abelian group. These connections offer a glimpse into the broader applicability and unifying power of module theory, even if the general definitions often feel like a descent into existential dread.

Definition

The formal definition of a finitely generated module is, predictably, rather precise. A left R-module M is finitely generated if there exists a finite collection of elements, let’s call them $a_1, a_2, ..., a_n$ , residing within M, such that any element $x$ in M can be expressed as a finite R-linear combination of these $a_i$ . That is, for any $x \in M$ , there exist corresponding scalars $r_1, r_2, ..., r_n$ from the ring R such that $x = r_1 a_1 + r_2 a_2 + ... + r_n a_n$ .

This particular set, $\{a_1, a_2, ..., a_n\}$ , is aptly named a generating set for M. Now, pay attention: a finite generating set, unlike a basis in a vector space, does not inherently need to be linearly independent over R. This distinction is crucial and often overlooked by those who prefer to keep things simple. A module can be finitely generated even if its generators are redundant.

What is true, and perhaps more illuminating, is that M is finitely generated if and only if there exists a surjective R-linear map (an epimorphism) from a free module of finite rank, $R^n$ , onto M. In more casual terms, this means M can be viewed as a "quotient" of a free module, specifically one that has a finite number of basis elements. This formulation often simplifies proofs and highlights the module's structure as a "constrained" version of a free module.

Consider the integers, $\mathbb{Z}$ , viewed as a $\mathbb{Z}$ -module. The singleton set $\{1\}$ is a perfectly good generating set. Any integer $x$ can be written as $x \cdot 1$ . Simple. However, the set of all prime numbers is also a generating set for $\mathbb{Z}$ (any integer can be written as a sum/difference of primes, by various theorems, though this is not a practical way to generate it). But this infinite set contains no finite generating set of minimal cardinality. You’d need at least two primes (e.g., 2 and 3) to generate all integers, but $\{1\}$ does it with one. So, while a set S might generate a finitely generated module, it doesn't guarantee that S itself is finite, or that it contains the most efficient finite generating set. It merely implies that some finite subset of S will suffice. The universe, it seems, isn't always efficient, but it does allow for efficiency when it matters.

In the special case where the module M happens to be a vector space over a field R, and the chosen generating set is also linearly independent, then the number $n$ (the size of the generating set) is uniquely determined. This $n$ is precisely what we call the dimension of M. This "well-defined" nature is a hallmark of vector spaces, a luxury not always afforded to general modules, as proven by the dimension theorem for vector spaces.

Every module, no matter how vast or unwieldy, can be understood as the union of the directed set of its finitely generated submodules. This is a rather profound statement, implying that even infinite modules are, in some sense, built up from their finite components.

A module M is finitely generated if and only if any increasing chain of its submodules, $M_i$ , whose union is M, must "stabilize." This means there must be some index $i$ where $M_i$ has already encompassed the entire module M. This condition, often paired with Zorn's lemma, implies that any nonzero finitely generated module must possess maximal submodules. If every increasing chain of submodules stabilizes – meaning every submodule within M is itself finitely generated – then M is bestowed with the grand title of a Noetherian module. It's a rather elegant connection, demonstrating how finite generation underpins broader structural properties.

Examples

Let's ground this theory with some concrete instances, to prevent your mind from drifting into purely abstract despair.

Cyclic Modules: If a module can be generated by a single, solitary element, it's deemed a cyclic module. These are the simplest of the finitely generated modules, a sort of minimalist masterpiece in module theory.
Fractional Ideals: Consider an integral domain R, and let K be its field of fractions. Any finitely generated R-submodule I of K turns out to be a fractional ideal. This means you can always find some non-zero element $r$ in R such that $rI$ is neatly contained within R. The "trick" is to simply take $r$ as the product of the denominators of the finite set of generators of I. If R is a Noetherian ring, this relationship becomes even stronger: every fractional ideal must arise in this manner.
Finitely Generated Abelian Groups: As previously mentioned, finitely generated modules over the ring of integers, $\mathbb{Z}$ , are precisely the finitely generated abelian groups. These groups are remarkably well-understood, thanks to the powerful structure theorem for finitely generated modules over a principal ideal domain, with $\mathbb{Z}$ serving as a prime example of a principal ideal domain. This theorem provides a complete classification, which is a rare and beautiful thing in mathematics.
Vector Spaces over Division Rings: When the underlying ring is a division ring (a ring where every non-zero element has a multiplicative inverse, like a field, but multiplication might not be commutative), the finitely generated (say, left) modules are nothing more than finite-dimensional vector spaces over that division ring. Again, the finite generation condition simply translates to finite dimensionality in these well-behaved contexts.

Some Facts

Now for the inevitable truths, some more palatable than others.

Every homomorphic image of a finitely generated module is, reassuringly, also finitely generated. This means that if you map a finitely generated module onto another module using a structure-preserving map, the image won't suddenly become infinitely complex. It retains its "finitely constructible" nature. However, and this is where things get interesting, submodules of finitely generated modules are not necessarily finitely generated themselves. This is a common pitfall for the unwary.

Consider the ring $R = \mathbb{Z}[X_1, X_2, ...]$ — the ring of all polynomials with integer coefficients in a countably infinite number of variables $X_1, X_2, X_3, \dots$ . This ring R, as an R-module over itself, is finitely generated by the single element $\{1\}$ . Every polynomial in R is simply $p(X_1, X_2, \dots) \cdot 1$ . Now, let's look at the submodule K consisting of all polynomials in R that have a zero constant term. This submodule K is generated by the infinite set $\{X_1, X_2, X_3, \dots\}$ . To express any element in K, you'd potentially need an arbitrarily large, though always finite, number of these variables. However, no finite subset of $\{X_1, X_2, X_3, \dots\}$ can generate all of K, because any finite set of variables would only allow you to form polynomials involving those variables, leaving out all polynomials that depend on variables not in your chosen finite set. Thus, K is not finitely generated. This illustrates the subtle, yet significant, difference between a module being finitely generated and its submodules sharing that property.

In general, a module is dubbed Noetherian if every submodule within it is finitely generated. This is a much stronger condition. A finitely generated module over a Noetherian ring is, in fact, a Noetherian module itself. This property is so fundamental that it actually characterizes Noetherian rings: a module over a Noetherian ring is finitely generated if and only if it is a Noetherian module. This bears a resemblance to, but is not identical to, Hilbert's basis theorem, which states that if R is a Noetherian ring, then the polynomial ring $R[X]$ (in a single variable) is also Noetherian. Both theorems, however, underscore the idea that "finiteness" properties tend to propagate nicely in Noetherian settings. These facts further imply that any finitely generated commutative algebra over a Noetherian ring will also turn out to be a Noetherian ring.

More broadly, an algebra (which could be a ring itself) that is a finitely generated module over a base ring is automatically a finitely generated algebra. The converse also holds under specific conditions: if a finitely generated algebra is also "integral" over its coefficient ring, then it must be a finitely generated module. (One might want to consult integral element for the precise definition of "integral.")

Consider an exact sequence of modules: $0 \to M' \to M \to M'' \to 0$ . This sequence represents a particular relationship between three modules. If both $M'$ and $M''$ are finitely generated, then M itself is guaranteed to be finitely generated. This is a convenient property for "gluing together" finitely generated modules. There are also partial converses, which is mathematics' way of saying "it's not always that simple." If M is finitely generated and $M''$ is finitely presented (a condition stronger than merely finitely generated, as we'll discuss), then $M'$ must also be finitely generated. Furthermore, M is Noetherian (or Artinian) if and only if both $M'$ and $M''$ are Noetherian (or Artinian, respectively).

Finally, let B be a ring and A its subring, such that B is a faithfully flat right A-module. Under these rather specific conditions, a left A-module F is finitely generated (or finitely presented) if and only if the B-module $B \otimes_A F$ (the tensor product) is also finitely generated (or finitely presented). This property, while technical, highlights how "finiteness" can be preserved under certain extensions of the base ring.

Finitely Generated Modules over a Commutative Ring

When we restrict our attention to finitely generated modules over a commutative ring R, certain powerful tools come into play. Nakayama's lemma is, without exaggeration, fundamental here. It's a cornerstone that often allows mathematicians to translate properties known from finite-dimensional vector spaces into the more general context of finitely generated modules.

For instance, a rather elegant consequence of Nakayama's lemma is this: if $f: M \to M$ is a surjective R-endomorphism (a homomorphism from M to itself) of a finitely generated module M, then $f$ must also be injective. This means $f$ is, in fact, an automorphism of M. In plainer terms, if you can map a finitely generated module onto itself, you're essentially just rearranging its elements without losing any distinctness. Such a module M is then called a Hopfian module. Dually, an Artinian module M (one where every descending chain of submodules stabilizes) exhibits a similar property, being coHopfian: any injective endomorphism $f$ on M must also be surjective. The mathematical universe, it seems, enjoys a good symmetry, even if it's often hidden. The Forster–Swan theorem provides another useful result, giving an upper bound for the minimal number of generators required for a finitely generated module M over a commutative Noetherian ring. Because sometimes, knowing how many you need is half the battle.

Any R-module, regardless of its size, can be expressed as an inductive limit of its finitely generated R-submodules. This is a powerful conceptual tool, as it often allows us to prove properties for general modules by first establishing them for the "simpler" finitely generated case. This is particularly useful for weakening assumptions, such as in the characterization of flatness using the Tor functor.

A critical link between finite generation and integral elements emerges in the study of commutative algebras. To say that a commutative algebra A is a "finitely generated ring" over R means there's a finite set of elements $G = \{x_1, \dots, x_n\}$ in A such that the smallest subring of A containing both G and R is A itself. Note the distinction: "finitely generated as a ring" allows for arbitrary products of generators, not just R-linear combinations. For example, a polynomial ring $R[x]$ is finitely generated by $\{1, x\}$ as a ring, but it is not finitely generated as an R-module (unless $R$ is the zero ring or $x$ is nilpotent over $R$ ). The elements $\{1, x, x^2, x^3, \dots\}$ are all distinct and linearly independent over $R$ (assuming $R$ is non-zero).

However, a crucial equivalence exists: if A is a commutative algebra (with unity) over R, then the following two statements are equivalent:

A is a finitely generated R-module.
A is both a finitely generated ring over R and an integral extension of R. This equivalence highlights that the "integrality" condition bridges the gap between these two notions of finite generation, providing a more structured and manageable environment.

Generic Rank

Let M be a finitely generated module over an integral domain A, and let K be its field of fractions. The quantity $\operatorname{dim}_K(M \otimes_A K)$ is known as the generic rank of M over A. This value, in essence, tells you the "dimension" of the module if you were to "localize" it over the field of fractions, treating A-scalars as K-scalars. This number precisely corresponds to the maximum number of A-linearly independent vectors one can find in M. Equivalently, it is the rank of a maximal free submodule of M, similar to the rank of an abelian group. Since the quotient module $(M/F)_{(0)}$ (the localization at the zero prime ideal) is zero, $M/F$ itself must be a torsion module.

When A is a Noetherian ring, we benefit from the principle of generic freeness. This theorem guarantees the existence of a specific element $f$ (which depends on the module M) within A such that the localized module $M[f^{-1}]$ becomes a free $A[f^{-1}]$ -module. The rank of this resulting free module is exactly the generic rank of M.

Now, let's consider a scenario where the integral domain A is an $\mathbb{N}$ -graded algebra over a field k, generated by a finite number of homogeneous elements with degrees $d_i$ . If M is also a graded module, we can define its Poincaré series as $P_M(t) = \sum (\operatorname{dim}_k M_n)t^n$ . By the venerable Hilbert–Serre theorem, there exists a polynomial F such that $P_M(t) = F(t)\prod (1-t^{d_i})^{-1}$ . In this context, the value $F(1)$ surprisingly yields the generic rank of M. It's almost as if the polynomials are telling us secrets about the module's structure.

A truly remarkable result for modules over a principal ideal domain (PID) is that a finitely generated module is torsion-free if and only if it is free. This is a direct consequence of the powerful structure theorem for finitely generated modules over a principal ideal domain, which states that any such module can be decomposed into a direct sum of a torsion module and a free module. Alternatively, one can prove this directly: let M be a torsion-free finitely generated module over a PID A, and let F be a maximal free submodule. Since M is finitely generated, we can find an element $f \in A$ (e.g., the product of denominators) such that $fM \subset F$ . Because F is free and A is a PID, its submodule $fM$ must also be free. The map $f: M \to fM$ is an isomorphism because M is torsion-free (if $fm=0$ , then $m=0$ ). Hence, M must be free.

Extending this, a finitely generated module over a Dedekind domain A (or, more generally, a semi-hereditary ring) is torsion-free if and only if it is projective. Consequently, any finitely generated module over a Dedekind domain can be expressed as a direct sum of a torsion module and a projective module. The generic rank of such a module over A then simply corresponds to the rank of its projective component. It's a neat way to break down complex structures into more manageable parts.

Equivalent Definitions and Finitely Cogenerated Modules

Beyond the primary definition, mathematicians, in their pursuit of clarity (or perhaps just more ways to say the same thing), have found several equivalent conditions for a module M to be finitely generated (f.g.). These can sometimes offer alternative perspectives or prove more useful in specific contexts:

For any arbitrary family of submodules $\{N_i \mid i \in I\}$ in M, if their sum $\sum_{i \in I} N_i = M$ , then it must be possible to find a finite subset F of I such that $\sum_{i \in F} N_i = M$ . This means M can't be "infinitely generated" by sums.
For any chain of submodules $\{N_i \mid i \in I\}$ in M, if their union $\bigcup_{i \in I} N_i = M$ , then some single $N_i$ in that chain must already be equal to M. The module can't be the limit of an infinitely ascending chain without eventually becoming that chain itself.
If $\phi: \bigoplus_{i \in I} R \to M$ is an epimorphism (a surjective homomorphism) from an infinite direct sum of copies of R onto M, then there must exist a finite subset F of I such that the restricted map $\phi: \bigoplus_{i \in F} R \to M$ is still an epimorphism. This implies that M only needs a finite number of "inputs" from a free module to be generated.

It's evident from these conditions that being finitely generated is a property that is preserved under Morita equivalence, which is a fancy way of saying that modules that are "the same" in a categorical sense will share this property.

These conditions also provide a convenient springboard to define a dual concept: a finitely cogenerated module M (f.cog.). While finitely generated modules are "built up" from a finite set, finitely cogenerated modules are "cut down" to nothing by a finite set of submodules. The following conditions are equivalent for a module to be finitely cogenerated:

For any family of submodules $\{N_i \mid i \in I\}$ in M, if their intersection $\bigcap_{i \in I} N_i = \{0\}$ (the zero module), then there must exist a finite subset F of I such that $\bigcap_{i \in F} N_i = \{0\}$ . This is the dual of the sum condition for f.g. modules.
For any chain of submodules $\{N_i \mid i \in I\}$ in M, if their intersection $\bigcap_{i \in I} N_i = \{0\}$ , then some single $N_i$ in that chain must already be equal to $\{0\}$ . The module can't be infinitely "cut down" without eventually disappearing.
If $\phi: M \to \prod_{i \in I} N_i$ is a monomorphism (an injective homomorphism) into an infinite direct product of R-modules $N_i$ , then there must exist a finite subset F of I such that the restricted map $\phi: M \to \prod_{i \in F} N_i$ is still a monomorphism. This means M can be faithfully embedded into a finite product of modules.

Both finitely generated and finitely cogenerated modules hold intriguing relationships with Noetherian and Artinian modules, as well as with the Jacobson radical $J(M)$ and the socle $\operatorname{soc}(M)$ of a module. These facts beautifully illustrate the duality between the two conditions:

M is Noetherian if and only if every submodule N of M is finitely generated. This means every part of a Noetherian module is "finitely constructible."
M is Artinian if and only if every quotient module M/N is finitely cogenerated. This means every "slice" of an Artinian module is "finitely decomposable."
M is finitely generated if and only if $J(M)$ (the Jacobson radical, representing "small" elements) is a superfluous submodule of M, and the quotient module $M/J(M)$ is finitely generated. This suggests that if you "mod out" the radical, the module remains finitely generated.
M is finitely cogenerated if and only if $\operatorname{soc}(M)$ (the socle, representing "minimal" elements) is an essential submodule of M, and $\operatorname{soc}(M)$ is finitely generated. This suggests that if the "minimal" part is finitely generated and pervasive, the whole module is finitely cogenerated.
If M is a semisimple module (meaning it's a direct sum of simple modules, like the socle of any module N), then it is finitely generated if and only if it is finitely cogenerated. In these "nicest" of modules, the two concepts merge.
If M is finitely generated and non-zero, it is guaranteed to possess a maximal submodule, and any quotient module $M/N$ will also be finitely generated. Finitely generated modules propagate their "finiteness" upwards through quotients.
If M is finitely cogenerated and non-zero, it is guaranteed to possess a minimal submodule, and any submodule N of M will also be finitely cogenerated. Finitely cogenerated modules propagate their "finiteness" downwards through submodules.
If both N and $M/N$ are finitely generated, then M itself is finitely generated. The same holds true if "f.g." is replaced with "f.cog." This is a useful property for building up modules.

Finitely cogenerated modules are required to have finite uniform dimension. This is intuitively clear if you consider the characterization involving the finitely generated essential socle. However, somewhat asymmetrically, finitely generated modules do not necessarily have finite uniform dimension. For instance, consider an infinite direct product of non-zero rings. This structure forms a finitely generated (specifically, a cyclic!) module over itself, yet it clearly contains an infinite direct sum of non-zero submodules, implying an infinite uniform dimension. Similarly, finitely generated modules don't necessarily have finite co-uniform dimension either; any ring R with unity where $R/J(R)$ is not a semisimple ring serves as a counterexample. The world, it seems, is rarely perfectly symmetrical.

Finitely Presented, Finitely Related, and Coherent Modules

To refine our understanding of module structure even further, we introduce more stringent conditions. Recall that a finitely generated module M is one for which there exists an epimorphism (a surjective R-linear map) from a free module $R^k$ onto M: $f: R^k \to M$ .

Now, let's suppose we have an epimorphism $\varphi: F \to M$ for a module M and some free module F.

Finitely Related Module: If the kernel of $\varphi$ (the set of elements in F that map to zero in M) is finitely generated, then M is termed a finitely related module. Since M is isomorphic to $F/\operatorname{ker}(\varphi)$ , this definition essentially states that M is constructed by taking a free module and imposing only a finite number of "relations" among its generators (these relations are precisely the generators of $\operatorname{ker}(\varphi)$ ).
Finitely Presented Module: If the kernel of $\varphi$ is finitely generated and the free module F itself has finite rank (i.e., $F = R^k$ for some finite $k$ ), then M is said to be a finitely presented module. This is a particularly well-behaved class of modules, as it means M can be fully described using a finite number of generators (the images of the $k$ generators of $R^k$ ) and a finite number of relations (the generators of $\operatorname{ker}(\varphi)$ ). This makes them, in a sense, "finitely describable" from both ends. You can think of this as a free presentation where both the domain and the kernel are finitely generated. Finitely presented modules have an elegant abstract characterization: they are precisely the compact objects within the category of R-modules.
Coherent Module: A coherent module M is a finitely generated module with the additional property that all of its finitely generated submodules are, in fact, finitely presented. This is a powerful condition, implying a certain internal consistency and regularity within the module's structure.

Over any ring R, the relationships between these concepts form a clear hierarchy: coherent modules are always finitely presented, and finitely presented modules are, by definition, both finitely generated and finitely related. However, for a Noetherian ring R – a ring where every ideal is finitely generated, and thus every submodule of a finitely generated module is finitely generated – these distinctions collapse. In this more structured environment, the conditions of being finitely generated, finitely presented, and coherent become entirely equivalent for any given module. Ah, simplicity, if only it were universal.

Some interesting crossovers occur when considering projective or flat modules. For instance, any finitely generated projective module is automatically finitely presented. Furthermore, a finitely related flat module will always be projective. These are the kinds of connections that make module theory both frustrating and fascinating.

It is also true that the following conditions are equivalent for a ring R:

R is a right coherent ring.
The module $R_R$ (R viewed as a right module over itself) is a coherent module.
Every finitely presented right R-module is coherent. These equivalences demonstrate how the properties of modules can reflect back onto the properties of the ring itself.

Although the definition of coherence might seem more cumbersome than merely being finitely generated or finitely presented, it offers a distinct advantage: the category of coherent modules forms an abelian category. This is a highly desirable property in category theory, meaning that concepts like kernels, cokernels, and images behave in a particularly nice, functorial way. In contrast, the categories of finitely generated or finitely presented modules do not, in general, form abelian categories. Sometimes, the more complex definition actually leads to a more elegant and useful theoretical framework.