Algebra over a field
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
Examples
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The continuous functions on $[0, 1]$, with pointwise addition, multiplication by real numbers, and pointwise multiplication
Undergraduate
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
Undergraduate
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
Things that may be mistaken for examples.
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The plane $\reals^2$, with vector addition and multiplication by real numbers
Undergraduate
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
Undergraduate
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 $F$.
Notes
Many authors require an algebra to be associative and to have an identity element $1$ (to be unital). Others, as here, also allow nonassociative algebras, such as Lie algebras. An algebra over a ring $R$ is defined the same way, with a module over $R$ in place of the vector space.
The word “algebra” has several other meanings: the branch of mathematics, a σ-algebra of sets, and, in universal algebra, every set with a collection of operations.
Distinctions involving this term
- 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 $F$.