Module (algebra)
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
Examples
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The plane $\reals^2$, with vector addition and multiplication by real numbers
Undergraduate
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
Undergraduate
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.
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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 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
Undergraduate
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 $M$
Graduate
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.
Notes
When the ring $R$ is not commutative, this definition gives a left module, and a right module multiplies by scalars on the right.
“Module” also names a file of code in programming languages such as Python, which is unrelated.
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 $F$.