Algebras and modules

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Super quick review of algebras, ideals, and modules.

A reference for this material is Jacobson, Nathan. Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp. ISBN:0-7167-1480-9. More tailored to smooth manifolds is Conlon Lawrence. Differentiable manifolds, a first course. Birkhäuser, 1993.

From now on, all our vector spaces are over $\mathbb {R}$ unless otherwise stated.

Algebras Let $A$ be a real vector space. We say $A$ is an algebra if there is an $\mathbb {R}$-bilinear associative product $A \times A \to A$.

$A$ is unital if it has a 1.

$A$ is commutative if $ab=ba$ for all $a,b \in A$.

From now on, all our algebras are unital unless otherwise stated.

$A = \mathbb {R}$.

Continuous functions If $X$ is a topological space, then $A = C(X)$ is an algebra.

Smooth functions If $M$ is a smooth manifold, then $A = C^{\infty }(M)$ is an algebra.

Polynomial algebra If $n \in \mathbb {N}$, then $A = \mathbb {R}[x_1, \ldots , x_n] $ is an algebra.

Matrices If $n \in \mathbb {N}$, then $A = M_n (\mathbb {R})$ is an algebra.

Linear endomorphisms If $V$ is a vector space, then

\[ A = End(V) : = \{ \, T : V \to V \, \vert \, T \text { is linear}\, \} \]

is an algebra.

Linear differential operators If $n\in \mathbb {R}$, then $\mathcal {D}_n$, the set of functions

\[ C^{\infty }(\mathbb {R}^n) \to C^{\infty } (\mathbb {R}^n) \]

generated by the identity and $\partial _1, \ldots , \partial _n$ under linear combinations and composition, is an algebra.

Tensor algebra If $V$ is a vector space, then

\[ A = \bigoplus _{j = 0 }^{\infty } V^{\otimes j} \]

is an algebra.

A non-unital algebra Let $A = C_c (\mathbb {R})$ the set of continuous functions $f : \mathbb {R} \to \mathbb {R}$ with compact support.

Ideal A proper subset $I \subset A$ is an ideal if it is a sub-vector space, and

\[ a b , \, ba \in I \, \text { for all } a \in A, b \in I . \]

We write $I \triangleleft A$ to denote that $I$ is an ideal.

For any subset $S \subset A$, there is a smallest ideal of $A$ containing $S$, which coinides with the intersection of all the ideals containing $S$. This ideal is also the set of all finite linear combinations of elements of the form $asb$ with $a,b \in A$ and $s \in S$.

Functions vanishing on a set Let $A = C(X)$ and $Y \subset X$ a nonempty subset. Then

\[ I = \{ f \in C(X) \, \vert \, f (x) = 0 \text { for all } x \in Y \} \]

is an ideal of $A$.

Polynomials of high degree If $A = \mathbb {R} [x_1, \ldots , x_n]$ and $k \in \mathbb {N}$, then the vector space of polynomials of degree $\geq k$ is an ideal of $A$.

Quotient algebra Let $A$ be an algebra and $I \triangleleft A$. The vector space $A/ I$ is also an algebra in a natural way.

Complex numbers Let $A = \mathbb {R}[x]$ and $I\triangleleft A$ the ideal generated by $( x^2 + 1 ) $. Then

\[ A / I = \mathbb {C} . \]

Modules Let $A$ be an algebra. An $A$-module is an abelian group $M$ that can be multiplied by elements of $A$ in an associative way. That is, there is a function $A \times M \to M$ such that

$(a_1a_2 )m = a_1 (a_2m)$ for all $a_1, a_2 \in A$ and $m \in M$.
$(a_1 + a_2)m = a_1 m + a_2 m$ for all $a_1, a_2 \in A$ and $m \in M$.
$a(m_1 + m_2 ) = a m_1 + a m_2$ for all $a \in A$ and $m_1, m_2 \in M$.
$1m = m $ for all $m \in M$.

Vector spaces If $A = \mathbb {R}$, then the $A$-modules are exactly the vector spaces.

The obvious one If $A$ is an algebra, then $A$ is an $A$-module.

$\mathfrak {X}(M)$ is a $C^{\infty }(M)$-module.

$C^{\infty }(\mathbb {R}^n )$ is a $\mathcal {D}_n$-module.

$\mathbb {R}^n$ is an $M_n(\mathbb {R})$-module.

Any vector space $V$ is an $End (V) $-module.

Module morphisms Let $M$ and $N$ be two $A$-modules. A module morphism is a linear map $T : M \to N$ such that $aT = Ta $ for all $a \in A$. We denote by $Hom_A(M,N)$ the set of module morphisms from $M$ to $N$. If $A$ is commutative, this is again an $A$-module.

If it is not expected to cause confusion, we write $Hom(M,N)$ instead of $Hom_A(M,N)$.

Dual $M^{\ast } : = Hom _A ( M , A).$

Submodule Let $M$ be an $A$-module and $N \subset M$. $N$ is a submodule if it is a sub-vector space and multiplying elements of $N$ by $A$ takes them within $N$. We write $N \leq _A M$ to denote the fact that $N$ is a submodule of $M$.

Span Let $M$ be an $A$-module and $S \subset M$. We denote by $span_A(S) \leq _A M $ to be one (or all) of the following:

The smallest submodule of $M$ that contains $S$.
The intersections of all submodules of $M$ containing $S$.
The set of linear combinations of elements of $S$ with coefficients in $A$.

$M$ is finitely generated if there is a finite set $S$ with $span _A(S) = M$.

Direct sum If $M$ and $N$ be $A$-modules. Then the direct sum $M \oplus N$ is an $A$-module.

Quotient If $M$ is an $A$-module and $N\leq _A M$, the quotient vector space $M / N$ is also an $A$-module in a natural way.