List of Modules

Introduction

This article lists the modules and the vector spaces of the corpus, with the rank, the basis, the torsion and the freeness of each. A module over a ring generalises a vector space over a field, and the list records exactly which of the four notions survives the generalisation: every vector space is free and has a basis, a module need not be free, the rank is defined for free modules and for torsion-free modules over a domain, and torsion is a phenomenon a field does not have.

Every entry points to the article that introduces the module or the vector space and states its invariant. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a module that fails to have a basis or to be torsion-free where a vector space would not, with the failure named and the article that records it.

Vector Spaces

A vector space over a field $F$ is a module over $F$; because $F$ is a field, every module is free, every linearly independent set extends to a basis, and any two bases have the same cardinality, the dimension. Subspaces, quotients, direct sums, the dual and the space of linear maps are all vector spaces, and the dimension is additive in short exact sequences.

Vector space	Rank, basis, torsion, freeness	Introduced in
$F^n$	dimension $n$, the standard basis $e_1,\dots,e_n$; free, torsion-free	Vector Spaces
Arbitrary vector space $V$	a basis exists by Zorn's lemma; dimension $\dim_F V$; free	Vector Spaces
Subspace and quotient	subspaces are direct summands; $\dim V = \dim W + \dim V/W$ for a subspace $W$	Vector Spaces
Direct sum $\bigoplus_i V_i$	dimension the sum of the dimensions; free	Vector Spaces
Dual space $V^* = \operatorname{Hom}_F(V,F)$	dimension $\dim_F V$ in finite dimension; free	Multilinear Spaces
Space of linear maps $\operatorname{Hom}_F(V,W)$	dimension $\dim_F V \cdot \dim_F W$; free	Vector Spaces
The vector space $\mathbb{H}^n$ over the division ring $\mathbb{H}$	a free module of rank $n$, of real dimension $4n$; every $\mathbb{H}$-module is free	Quaternion Ideals and Simplicity
Non-example: a torsion element in a vector space	does not occur: a nonzero scalar multiple of a nonzero vector is nonzero, so $T(V) = 0$	Vector Spaces
Non-example: a module over a field that is not free	does not occur: over a field every module is free, by the existence of bases	Vector Spaces

Free, Projective and Injective Modules

A module is free when it has a basis, projective when it is a direct summand of a free module, and injective when it is a direct summand of every module containing it; free implies projective, and over a principal ideal domain projective, free and torsion-free coincide for finitely generated modules. The rank is the cardinality of a basis of a free module, well defined by the invariant basis number property.

Module	Rank, basis, torsion, freeness	Introduced in
Free module $R^{(I)}$	basis $\{e_i\}$; free by definition, rank $\|I\|$	Modules
Finitely generated free module $R^n$	rank $n$; free, torsion-free over a domain	Direct Sums, Free Modules and Rank
Projective module $P$	a direct summand of a free module; not necessarily free	Projective and Injective Modules
Injective module $I$	a direct summand of every containing module; Baer's criterion tests it	Projective and Injective Modules
$\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$	divisible abelian groups; injective as $\mathbb{Z}$-modules	Projective and Injective Modules
$R/(a)$ over a PID	not free for $a \neq 0$; torsion, of projective dimension $1$	Projective and Injective Modules
$\mathbb{Z}[1/p]$	a localisation of $\mathbb{Z}$; torsion-free, not finitely generated, not free	Projective and Injective Modules
Free resolution $L_1 \to L_0 \to M \to 0$	exhibits $M$ as a quotient of free modules; measures the failure of freeness	Direct Sums, Free Modules and Rank
Non-example: a projective module that is not free	fails freeness over a general ring: projective is necessary but not sufficient	Projective and Injective Modules
Non-example: the invariant basis number property	fails over a general ring: the rank is well defined only when IBN holds	Direct Sums, Free Modules and Rank

Finitely Generated Modules over a Principal Ideal Domain

Over a principal ideal domain every submodule of a free module is free, every finitely generated torsion-free module is free, and every finitely generated module is a direct sum of a free module and cyclic torsion modules with unique invariants. The torsion submodule $M_{\mathrm{tor}}$ collects the elements annihilated by a nonzero element of the ring, the rank is the free rank, and the theorem specialises to the classification of finitely generated abelian groups and to the rational and Jordan forms.

Module	Rank, basis, torsion, freeness	Introduced in
$M \cong R^r \oplus R/(d_1) \oplus \cdots \oplus R/(d_k)$	free rank $r$, invariant factors $d_1 \mid \cdots \mid d_k$; torsion the second summand	Modules over a PID
Torsion submodule $M_{\mathrm{tor}}$	the elements of nonzero annihilator; a direct summand of a finitely generated module	Modules over a PID
$p$-primary component $M_p$	the torsion part at the prime $p$; elementary divisors $p^{e_{p,j}}$	Modules over a PID
Cyclic module $R/(a)$	rank $0$; torsion; free exactly when $a = 0$	Modules over a PID
$\mathbb{Z}$-module of finite type	the case $R = \mathbb{Z}$: free part $\mathbb{Z}^r$ plus torsion	Finitely Generated Abelian Groups
$k[x]$-module	the case $R = k[x]$: the rational and Jordan forms of an operator	Modules over $k[x]$ and the Jordan Form
Torsion-free finitely generated module over a PID	free, of rank equal to the free rank	Modules over a PID
Non-example: $\mathbb{Q}$ as a $\mathbb{Z}$-module	torsion-free but not free: it is not finitely generated	Modules
Non-example: a submodule of a free module over a general ring	fails to be free: the statement needs a principal ideal domain	Modules

Torsion, Simple and Semisimple Modules

A module is simple when it has no proper nonzero submodule, semisimple when it is a direct sum of simples, and semisimple modules have no torsion in the module-theoretic sense over a division ring. The Jacobson radical $J(R)$ is the obstruction to semisimplicity, the density theorem describes the action on a semisimple module, and Schur's lemma makes the endomorphism ring of a simple module a division ring.

Module	Rank, basis, torsion, freeness	Introduced in
Simple module $S$	no proper nonzero submodule; $\operatorname{End}_A(S)$ a division ring	Simple and Semisimple Modules
Semisimple module	a direct sum of simples; every submodule a direct summand	Simple and Semisimple Modules
Isotypic component	the summands of one isomorphism class; endomorphism ring a matrix ring	Simple and Semisimple Modules
Jacobson radical $J(R)$	the intersection of the maximal left ideals; zero exactly when $R$ is semisimple	Simple and Semisimple Modules
Modules over a semisimple algebra	all semisimple; a direct sum of copies of the simple modules	Simple and Semisimple Modules
Non-example: the module $k[x]/(x^2)$ over $k[x]/(x^2)$	fails semisimplicity: it has a nonzero radical	Simple and Semisimple Modules
Non-example: a module of infinite length	fails the finite composition series: the length is not defined	Simple and Semisimple Modules

Modules over Division Rings and the Number Systems

Over a division ring every module is free, and the rank theory is that of vector spaces; the corpus develops this for $\mathbb{H}$ and for the biquaternions $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H} \cong M_2(\mathbb{C})$, where the zero divisors of $\mathbb{B}$ prevent a basis and the module theory is governed instead by Morita equivalence with complex vector spaces.

Module	Rank, basis, torsion, freeness	Introduced in
$\mathbb{H}^n$	free of rank $n$ over the division ring $\mathbb{H}$; basis $e_1,\dots,e_n$; real dimension $4n$	Quaternion Ideals and Simplicity
Finitely generated $\mathbb{B}$-module	isomorphic to $S^{\oplus k}$ for the unique simple $S = \mathbb{C}^2$; complex dimension $2k$	Modules over the Biquaternion Algebra
The defining module $S$ of $\mathbb{B}$	the unique simple $\mathbb{B}$-module; not free over $\mathbb{B}$ in the vector-space sense	Modules over the Biquaternion Algebra
$\mathbb{H}$-module structure	complex structure $I$ and quaternionic structure $\mathcal{J}$; $J(\mathbb{H}) = 0$	Quaternion Ideals and Simplicity
Lipschitz order $\mathbb{Z}\{e_0,e_1,e_2,e_3\}$	a torsion-free $\mathbb{Z}$-module of rank $4$, not free over a non-commutative order	Lattices and the Quaternion Lattice
Non-example: a $\mathbb{B}$-module treated as a vector space	fails to have a basis over $\mathbb{B}$: the ring has zero divisors	Biquaternion Algebra
Non-example: a torsion element of an $\mathbb{H}$-module	does not occur: $\mathbb{H}$ is a division ring, so every nonzero module is torsion-free	Quaternion Ideals and Simplicity

Homomorphisms, Duals and Tensor Products

The module $\operatorname{Hom}_R(M,N)$ of homomorphisms and the dual $M^* = \operatorname{Hom}_R(M,R)$ are again modules; the tensor product $M \otimes_R N$ is a module with the universal property of bilinear maps. The list records these constructions because they are the ones that distinguish the free and projective modules from the general ones: $\operatorname{Hom}$ and the tensor product are not exact in both variables, and the derived functors $\operatorname{Ext}$ and $\operatorname{Tor}$ measure the failure.

Construction	Rank, basis, torsion, freeness	Introduced in
$\operatorname{Hom}_R(M,N)$	a module; left exact in $M$; a free module when $M$ is free of finite rank	Modules
Dual module $M^* = \operatorname{Hom}_R(M,R)$	the linear forms; for a finitely generated projective module, $(M^)^ \cong M$	Projective and Injective Modules
Tensor product $M \otimes_R N$	right exact; $R \otimes_R M \cong M$; $(R^{(I)})\otimes_R N \cong N^{(I)}$	Modules
Tensor product of vector spaces	$\dim_F(V\otimes_F W) = \dim_F V \cdot \dim_F W$; free	Multilinear Spaces
Bilinear and multilinear maps	the universal property identifying them with maps on the tensor product	Multilinear Spaces
$\operatorname{Ext}_R^n(M,N)$ and $\operatorname{Tor}_n^R(M,N)$	the derived functors measuring the failure of exactness	Ext and Tor
Non-example: exactness of $\operatorname{Hom}_R(M,-)$	fails in general: it is left exact only, and $\operatorname{Ext}^1$ is the obstruction	Ext and Tor
Non-example: exactness of $-\otimes_R N$	fails in general: it is right exact only, and $\operatorname{Tor}_1$ is the obstruction	Ext and Tor

Summary

The list gathers the modules and vector spaces of the corpus. The vector spaces over a field are free, have bases and dimensions, have no torsion, and their subspaces and quotients are again vector spaces. The modules over a general ring are the free, projective and injective modules, with the rank defined by a basis under the invariant basis number property and the free resolutions measuring the failure of freeness. Over a principal ideal domain the finitely generated modules are classified by a free rank and the torsion invariants, with the cases $\mathbb{Z}$ and $k[x]$ giving the abelian groups and the Jordan form. The simple and semisimple modules, the Jacobson radical and the density theorem describe the semisimple side. Over a division ring every module is free; over the quaternions the rank theory is that of vector spaces, and over the biquaternions the module theory is the Morita equivalence with complex vector spaces. The non-examples — a torsion element of a vector space, a non-free module over a field, a projective module that is not free, a ring without the invariant basis number property, $\mathbb{Q}$ as a torsion-free non-free $\mathbb{Z}$-module, a submodule of a free module over a general ring, the module $k[x]/(x^2)$ and a module of infinite length, a $\mathbb{B}$-module treated as a vector space and a torsion element of an $\mathbb{H}$-module — each name the property that fails.

Summary of Notation

The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$V$, $W$, $\dim_F V$	vector spaces and their dimension
$V^*$	dual space
$R^{(I)}$, $R^n$	free module, finitely generated free module
$\operatorname{rk}_R L$	rank of a free module
$M_{\mathrm{tor}}$, $M_p$	torsion submodule and primary component
$R/(a)$	cyclic module
$J(R)$	Jacobson radical
$S$, $\operatorname{End}_A(S)$	simple module and its endomorphism division ring
$\mathbb{H}^n$, $S^{\oplus k}$	free $\mathbb{H}$-module, finitely generated $\mathbb{B}$-module
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	biquaternions
$\mathbb{Z}$, $\mathbb{Q}$	the integers and the rationals
$R^r \oplus R/(d_1)\oplus\cdots$	structure theorem over a PID