Vector Spaces : A General Introduction

Introduction

This article introduces the theory of vector spaces over a commutative ring. The classical theory of vector spaces assumes that the scalars form a field. But many of the constructions and theorems carry over to the more general setting where the scalars form a commutative ring. The resulting objects are called modules, and they are the natural generalization of vector spaces.

The treatment is introductory and purely mathematical. We assume familiarity with rings, fields, and the basic theory of vector spaces over a field.

We begin with the definition of a module over a commutative ring, then introduce the notions of submodules, homomorphisms, free modules, bases, and dimension. We compare the theory over a ring with the classical theory over a field, and we indicate where the two diverge.

Part I: Modules over a Commutative Ring

1. Definition of a Module

Let $R$ be a commutative ring with identity. An $R$-module is an abelian group $(M, +)$ equipped with a scalar multiplication

$$ R \times M \to M, \qquad (r, m) \mapsto r m, $$

satisfying the following axioms for all $r, s \in R$ and $m, n \in M$:

(M1) $r(m + n) = r m + r n$.

(M2) $(r + s)m = r m + s m$.

(M3) $(r s)m = r(s m)$.

(M4) $1 m = m$.

If $R$ is a field, an $R$-module is exactly a vector space over $R$. So modules generalize vector spaces to arbitrary commutative rings.

2. Elementary Properties

Let $M$ be an $R$-module. The following properties follow from the axioms.

(a) Multiplication by the zero scalar. For all $m \in M$,

$$ 0 m = 0. $$

(b) Multiplication by the zero element. For all $r \in R$,

$$ r 0 = 0. $$

(c) Sign rules. For all $r \in R$ and $m \in M$,

$$ (-r) m = r(-m) = -(r m), \qquad (-r)(-m) = r m. $$

(d) Distributivity over subtraction. For all $r, s \in R$ and $m, n \in M$,

$$ r(m - n) = r m - r n, \qquad (r - s)m = r m - s m. $$

3. Examples of Modules

(a) The ring itself. Every commutative ring $R$ is an $R$-module, with scalar multiplication given by the ring multiplication.

(b) The free module $R^n$. The set of $n$-tuples of elements of $R$,

$$ R^n = \{(r_1, \ldots, r_n) : r_i \in R\}, $$

is an $R$-module with componentwise addition and scalar multiplication.

(c) Ideals. Every ideal $I \subseteq R$ is an $R$-module.

(d) Quotient rings. If $I \subseteq R$ is an ideal, then $R/I$ is an $R$-module.

(e) The polynomial ring. The polynomial ring $R[x]$ is an $R$-module.

(f) The matrix ring. The set $M_n(R)$ of $n \times n$ matrices over $R$ is an $R$-module.

(g) Abelian groups as $\mathbb{Z}$-modules. Every abelian group is a $\mathbb{Z}$-module, with scalar multiplication given by repeated addition.

Part II: Submodules and Homomorphisms

4. Submodules

A subset $N \subseteq M$ of an $R$-module $M$ is a submodule if:

(S1) $N$ is a subgroup of $(M, +)$: for all $m, n \in N$, $m - n \in N$.

(S2) $N$ is closed under scalar multiplication: for all $r \in R$ and $m \in N$, $r m \in N$.

A submodule of a module over a field is exactly a vector subspace.

5. Module Homomorphisms

Let $M$ and $N$ be $R$-modules. An $R$-module homomorphism is a function

$$ \varphi : M \to N $$

such that for all $r \in R$ and $m, n \in M$:

(H1) $\varphi(m + n) = \varphi(m) + \varphi(n)$.

(H2) $\varphi(r m) = r \varphi(m)$.

An $R$-module homomorphism that is bijective is an $R$-module isomorphism. If there is an $R$-module isomorphism $M \to N$, we write $M \cong N$ and say that $M$ and $N$ are isomorphic.

The kernel of an $R$-module homomorphism $\varphi : M \to N$ is

$$ \ker \varphi = \{m \in M : \varphi(m) = 0\}. $$

The image of $\varphi$ is

$$ \operatorname{im} \varphi = \{\varphi(m) : m \in M\}. $$

The kernel is a submodule of $M$, and the image is a submodule of $N$.

6. Quotient Modules

Let $M$ be an $R$-module and $N \subseteq M$ a submodule. The quotient module $M/N$ is the set of cosets

$$ M/N = \{m + N : m \in M\} $$

with addition and scalar multiplication defined by

$$ (m + N) + (n + N) = (m + n) + N, \qquad r(m + N) = r m + N. $$

These operations are well-defined, and $M/N$ is an $R$-module.

The first isomorphism theorem states that if $\varphi : M \to N$ is an $R$-module homomorphism, then

$$ M / \ker \varphi \cong \operatorname{im} \varphi. $$

Part III: Free Modules and Bases

7. Linear Independence

Let $M$ be an $R$-module. A subset $S \subseteq M$ is linearly independent if for every finite subset $\{m_1, \ldots, m_k\} \subseteq S$, the equation

$$ r_1 m_1 + \cdots + r_k m_k = 0 $$

with $r_i \in R$ implies $r_1 = \cdots = r_k = 0$.

A subset $S \subseteq M$ spans $M$ if every element of $M$ can be written as a finite linear combination of elements of $S$.

A basis of $M$ is a linearly independent spanning set.

8. Free Modules

An $R$-module $M$ is free if it has a basis. If $M$ has a basis of cardinality $n$, then $M \cong R^n$.

Examples.

$R^n$ is a free $R$-module of rank $n$.
Every vector space over a field is free.
Over $\mathbb{Z}$, the free modules are the groups $\mathbb{Z}^n$.
Over $\mathbb{D}$, the free modules are the modules $\mathbb{D}^n$.

Non-examples.

$\mathbb{Z}/n\mathbb{Z}$ is a $\mathbb{Z}$-module that is not free for $n \geq 2$.
Any ideal $I \subseteq R$ that is not principal is not free as an $R$-module in general.

9. Rank

If $M$ is a free $R$-module with a basis of cardinality $n$, we say that $M$ has rank $n$. Over a field, the rank is the dimension, and it is well-defined: every basis has the same cardinality.

Over a commutative ring with $1 \neq 0$, the rank of a free module is well-defined: every such ring has the invariant basis number property. This property can fail for non-commutative rings: there are non-commutative rings over which $R^m \cong R^n$ for $m \neq n$.

10. Torsion

An element $m \in M$ is a torsion element if there exists a nonzero $r \in R$ such that

$$ r m = 0. $$

If $R$ is an integral domain, the set of torsion elements of $M$ is a submodule, called the torsion submodule of $M$. Over a general commutative ring this can fail: if $r m = 0$ and $s n = 0$ with $r, s \neq 0$, then $r s (m + n) = 0$, but $r s$ may itself be $0$, in which case $m + n$ need not be torsion. For example, in $\mathbb{Z}/6\mathbb{Z}$ viewed as a module over itself, $2$ and $3$ are torsion, since $3 \cdot 2 = 0$ and $2 \cdot 3 = 0$, but $2 + 3 = 5$ is not, since $5$ is a unit.

A module is torsion-free if its only torsion element is $0$.

Over a field, every module is torsion-free. Over a general commutative ring, torsion is a new phenomenon.

Example. In $\mathbb{Z}/n\mathbb{Z}$ as a $\mathbb{Z}$-module, every element is torsion.

Part IV: Comparison with Vector Spaces

11. What Carries Over

Many of the basic notions of linear algebra carry over to modules over a commutative ring without change:

Submodules, quotient modules, and homomorphisms.
The first isomorphism theorem.
Direct sums and direct products.
Free modules and bases.
Linear independence and spanning sets.

12. What Does Not Carry Over

Several key properties of vector spaces fail for modules over a general commutative ring:

(a) Not every module is free. Over a field, every module is free. Over a general ring, this is false. For example, $\mathbb{Z}/n\mathbb{Z}$ is not a free $\mathbb{Z}$-module.

(b) Not every submodule is a direct summand. Over a field, every subspace of a vector space has a complement. Over a general ring, this is false.

(c) Rank and the invariant basis number property. Over a field, every basis has the same cardinality. Over a commutative ring with $1 \neq 0$ this remains true, since every such ring has the invariant basis number property; the failure $R^m \cong R^n$ with $m \neq n$ occurs only for non-commutative rings.

(d) Torsion may exist. Over a field, every module is torsion-free. Over a general ring, torsion is a new phenomenon.

(e) Invertibility is detected by units, not by non-zero determinants. Over a field, every linear map has a determinant, and invertibility is equivalent to nonzero determinant. Over a general ring, the determinant is still defined, but invertibility is equivalent to the determinant being a unit of the ring.

13. The Case of a Field

When $R = F$ is a field, the theory of $F$-modules is exactly the theory of vector spaces over $F$. Every module is free, every submodule is a direct summand, every basis has the same cardinality, and there is no torsion. The classification of finitely generated $F$-modules is trivial: every finitely generated $F$-module is isomorphic to $F^n$ for some $n$.

14. The Case of $\mathbb{Z}$

When $R = \mathbb{Z}$, the theory of $\mathbb{Z}$-modules is exactly the theory of abelian groups. Not every abelian group is free: for example, $\mathbb{Z}/n\mathbb{Z}$ is not free. The classification of finitely generated $\mathbb{Z}$-modules is the structure theorem for finitely generated abelian groups: every finitely generated abelian group is isomorphic to

$$ \mathbb{Z}^r \oplus \mathbb{Z}/n_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/n_k\mathbb{Z} $$

for some $r \geq 0$ and $n_1, \ldots, n_k \geq 2$.

15. The Case of $\mathbb{D}$

When $R = \mathbb{D}$, the split complex numbers, the theory of $\mathbb{D}$-modules is more subtle because $\mathbb{D}$ has zero divisors. A $\mathbb{D}$-module need not be free, and torsion can occur. But since $\mathbb{D} \cong \mathbb{R} \times \mathbb{R}$ as a ring, every $\mathbb{D}$-module decomposes as a pair of $\mathbb{R}$-vector spaces:

$$ M \cong M_1 \times M_2 $$

where $M_1$ and $M_2$ are real vector spaces. This reduces the theory of $\mathbb{D}$-modules to the theory of real vector spaces.

Part V: Summary

16. Summary of the Theory

Notion	Over a field $F$	Over a commutative ring $R$
Module	Vector space	Module
Submodule	Subspace	Submodule
Homomorphism	Linear map	$R$-linear map
Free module	Always free	Not always free
Basis	Always exists	May not exist
Rank	Well-defined	Well-defined if IBN holds
Torsion	None	Possible
Determinant	Defined	Defined, invertibility iff unit
Classification	Trivial for f.g. modules	Structure theorem for PIDs

17. The Role of Commutative Rings

Commutative rings are the natural setting for:

Algebraic geometry. The ring of functions on an algebraic variety is a commutative ring, and the modules over it are the sheaves on the variety.
Number theory. The ring of integers of a number field is a commutative ring, and the modules over it are the ideals and fractional ideals of the field.
The theory of quadratic forms over rings. Clifford algebras can be defined over any commutative ring, and their representation theory is the theory of modules over the Clifford algebra.
The theory of modules. Modules over commutative rings generalize vector spaces, and their classification is a central problem in commutative algebra.

Summary

The article develops the theory of vector spaces over a commutative ring. An $R$-module is an abelian group with a scalar multiplication by $R$, and the treatment begins by carrying the vocabulary of linear algebra into that setting: submodules, quotient modules, homomorphisms, linear independence, free modules, rank and torsion, with their elementary properties and the isomorphism theorems.

The second half separates what survives the passage from a field to a ring from what does not. Submodules, quotients, homomorphisms and the isomorphism theorems carry over unchanged; free modules need not exist, not every submodule is a direct summand, and rank is well defined only when the ring has the invariant basis property. The comparison with the field case is recorded together with the two cases the rest of the corpus uses: the ring $\mathbb{Z}$, where the theory is the theory of abelian groups and the structure theorem classifies the finitely generated ones, and the split complex algebra $\mathbb{D}$, where the zero divisors make torsion and non-freeness possible.

The article closes with the role of commutative rings in the wider corpus, among them algebraic geometry, number theory, and the theory of quadratic forms and Clifford algebras over rings.