Vector Space vs. Module vs. algebra
A module and an algebra each generalize a vector space, in different directions. A module allows its scalars to come from a ring instead of a field, but, like a vector space, a module has no multiplication of two elements. An algebra keeps its scalars in a field but adds a bilinear multiplication of elements. Every algebra over a field $F$ is a vector space over $F$, and so every algebra over $F$ is also a module over $F$.
Definitions
Level:
High school
A vector space is a set of objects, called vectors, that can be added to each other and multiplied by numbers, called scalars, following the usual rules of arithmetic, such as $u + v = v + u$ and $2 (u + v) = 2 u + 2 v$.
Undergraduate
A vector space over a field $F$ is a set $V$ with a vector addition $V \times V \to V$ and a scalar multiplication $F \times V \to V$ such that, for all $a, b \in F$ and $u, v \in V$,
- $V$ with addition is a commutative group,
- Scalar multiplication is distributive over addition: $a (u + v) = a u + a v$ and $(a + b) v = a v + b v$,
- $(a b) v = a (b v)$, and
- $1 v = v$.
The elements of $V$ are called vectors, and the elements of $F$ are called scalars.
Examples
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The plane $\reals^2$, with vector addition and multiplication by real numbers
The plane $\reals^2$, with vector addition and multiplication by real numbers, is a vector space over $\reals$.
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The continuous functions on $[0, 1]$, with pointwise addition, multiplication by real numbers, and pointwise multiplication
The continuous functions on $[0, 1]$ form a real vector space, because sums and real multiples of continuous functions are continuous.
- The set of polynomials of degree at most $2$
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The $2 \times 2$ real matrices, with matrix addition, multiplication by real numbers, and matrix multiplication
The $2 \times 2$ real matrices form a real vector space of dimension $4$, with entrywise addition and scalar multiplication by real numbers.
Non-examples
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The pairs of integers $\mathbb{Z}^2$, with addition and multiplication by integers
The pairs of integers $\mathbb{Z}^2$, with multiplication by integers, do not form a vector space, because the integers do not form a field.
Properties
- Every vector space has a basis, assuming the axiom of choice, and all bases of the same vector space have the same number of elements, which is the dimension of the vector space.
- In a vector space, multiplication of two vectors is undefined, but can be introduced as in an algebra or an inner product space.
Undergraduate (abstract algebra)
A module over a ring $R$ is a set $M$ with an addition $M \times M \to M$ and a scalar multiplication $R \times M \to M$ such that, for all $r, s \in R$ and $x, y \in M$,
- $M$ with addition is a commutative group,
- $r (x + y) = r x + r y$ and $(r + s) x = r x + s x$,
- $(r s) x = r (s x)$, and
- $1 x = x$.
A module is defined like a vector space, except that the scalars come from a ring instead of a field.
Examples
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The plane $\reals^2$, with vector addition and multiplication by real numbers
The plane $\reals^2$, with vector addition and multiplication by real numbers, is a module over $\reals$, because every vector space over a field is a module over that field.
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The pairs of integers $\mathbb{Z}^2$, with addition and multiplication by integers
The pairs of integers $\mathbb{Z}^2$, with addition and multiplication by integers, form a module over the ring $\mathbb{Z}$, because the rules of a vector space hold with integer scalars.
-
The continuous functions on $[0, 1]$, with pointwise addition, multiplication by real numbers, and pointwise multiplication
The continuous functions on $[0, 1]$ form a module over $\reals$, because they form a vector space over $\reals$.
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The $2 \times 2$ real matrices, with matrix addition, multiplication by real numbers, and matrix multiplication
The $2 \times 2$ real matrices form a module over $\reals$, because they form a vector space over $\reals$.
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The smooth vector fields on a manifold $M$, as a module over the ring of smooth functions on $M$
The smooth vector fields on a manifold $M$ form a module over the smooth functions on $M$, because the product $f X$ of a smooth function $f$ and a smooth vector field $X$ is a smooth vector field. The smooth functions form a ring but not a field, because a function that is $0$ somewhere has no multiplicative inverse.
Properties
- A module over a field is the same as a vector space over that field.
- Every commutative group $M$ is a module over the integers, with $n x = x + \dots + x$ ($n$ terms) for positive integers $n$.
- A module need not have a basis. In the integers modulo $2$, as a module over the integers, $2 x = 0$ for every $x$, so no nonempty set of elements is linearly independent.
Undergraduate (abstract algebra)
An algebra over a field $F$ is a vector space $A$ over $F$ with a multiplication $A \times A \to A$, written $u v$, that is bilinear.
That is, for all $a \in F$ and $u, v, w \in A$,
- $(u + v) w = u w + v w$,
- $u (v + w) = u v + u w$, and
- $(a u) v = a (u v) = u (a v)$.
Examples
-
The continuous functions on $[0, 1]$, with pointwise addition, multiplication by real numbers, and pointwise multiplication
The continuous functions on $[0, 1]$ form an algebra over $\reals$, because the pointwise product $(f g)(x) = f(x) g(x)$ of continuous functions is continuous and bilinear.
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The $2 \times 2$ real matrices, with matrix addition, multiplication by real numbers, and matrix multiplication
The $2 \times 2$ real matrices form an algebra over $\reals$, because matrix multiplication is bilinear: $(a A) B = a (A B) = A (a B)$, and matrix multiplication distributes over addition.
Non-examples
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The plane $\reals^2$, with vector addition and multiplication by real numbers
The plane $\reals^2$, with only vector addition and multiplication by real numbers, is not an algebra, because no multiplication of two vectors is given.
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The pairs of integers $\mathbb{Z}^2$, with addition and multiplication by integers
The pairs of integers $\mathbb{Z}^2$, with addition and multiplication by integers, do not form an algebra over a field, because the scalars do not form a field and no multiplication of two pairs is given.
Properties
- The multiplication of an algebra need not be associative or commutative. A Lie algebra, whose multiplication is the Lie bracket, is neither.
- An algebra whose multiplication is associative and has an identity element is a ring, and the scalar multiplication of that algebra satisfies $a u = (a 1) u$.
- Every algebra over $F$ is a module over $F$, because every algebra over $F$ is a vector space over $F$.
Examples
Notes
- An algebra over a ring $R$ is a module over $R$ with a bilinear multiplication, so an algebra over a ring combines both generalizations.
- Many authors require an algebra to be associative and unital; a Lie algebra is an algebra only in the broader sense.