dense ring of linear transformations

Let $D$ be a division ring and $V$ a vector space over $D$. Let $R$ be a subring of the ring of endomorphisms (linear transformations) ${\mathrm{End}}_D(V)$ of $V$. Then $R$ is said to be a dense ring of linear transformations (over $D$) if we are given

1. any positive integer $n$,

2. any set $\{v_1,\mathrm{\dots },v_n\}$ of linearly independent vectors in $V$, and

3. any set $\{w_i,\mathrm{\dots },w_n\}$ of arbitrary vectors in $V$,

then there exists an element $f\in R$ such that

$$f(v_i)=w_i\mathit{\hspace{1em}}\text{ for }i=1,\mathrm{\dots },n.$$

Note that the linear independence of the $v_i$’s is essential in insuring the existence of a linear transformation $f$. Otherwise, suppose $0=\sum d_i{v}_i$ where $d_1\ne 0$. Pick $w_i$’s so that they are linearly independent. Then $0=f(\sum d_i{v}_i)=\sum d_{i}f(v_i)=\sum d_i{w}_i$, contradicting the linear independence of the $w_i$’s.

The notion of “dense” comes from topology: if $V$ is given the discrete topology and ${\mathrm{End}}_D(V)$ the compact-open topology, then $R$ is dense in ${\mathrm{End}}_D(V)$ iff $R$ is a dense ring of linear transformations of $V$.

Proof.

First, assume that $R$ is a dense ring of linear transformations of $V$. Recall that the compact-open topology on ${\mathrm{End}}_D(V)$ has subbasis of the form $B(K,U):=\{f\mid f(K)\subseteq U\}$, where $U$ is open and $K$ is compact in $V$. Since $V$ is discrete, $K$ is finite. Now, pick a point $g\in {\mathrm{End}}_D(V)$ and let

$$B=\bigcup _{\alpha \in I}\bigcap _{i=1}^{n(\alpha )}B(K_{i\alpha },U_{i\alpha })$$

be a neighborhood of $g$, $I$ some index set. Then for some $\alpha \in I$, $g\in \bigcap B(K_{i\alpha },U_{i\alpha })$. This means that $g(K_{i\alpha })\subseteq U_{i\alpha }$ for all $i=1,\mathrm{\dots },n(\alpha )$. Since each $K_{i\alpha }$ is finite, so is $K:=\bigcup K_{i\alpha }$. After some re-indexing, let $\{v_1,\mathrm{\dots },v_n\}$ be a maximal linearly independent subset of $K$. Set $w_j=g(v_j)$, $j=1,\mathrm{\dots },n$. By assumption, there is an $f\in R$ such that $f(v_j)=w_j$, for all $j$. For any $v\in K$, $v$ is a linear combination of the $v_j$’s: $v=\sum d_j{v}_j$, $d_j\in D$. Then $f(v)=\sum d_{j}f(v_j)=\sum d_{j}g{(}_j)=g(v)\in U_{i\alpha }$ for some $i$. This shows that $f(K_{i\alpha })\subseteq U_{i\alpha }$ and we have $f\in \bigcap B(K_{i\alpha },U_{i\alpha })\subseteq B$.

Conversely, assume that the ring $R$ is a dense subset of the space ${\mathrm{End}}_D(V)$. Let $v_1,\mathrm{\dots },v_n$ be linearly independent, and $w_1,\mathrm{\dots },w_n$ be arbitrary vectors in $V$. Let $W$ be the subspace spanned by the $v_i$’s. Because the $v_i$’s are linearly independent, there exists a linear transformation $g$ such that $g(v_i)=w_i$ and $g(v)=0$ for $v\notin W$. Let $K_i=\{v_i\}$ and $U_i=\{w_i\}$. Then the $K_i$’s are compact and the $U_i$’s are open in the discrete space $V$. Clearly $g\in \{h\mid h(v_i)=w_i\}=B(K_i,U_i)$ for each $i=1,\mathrm{\dots },n$. So $g$ lies in the neighborhood $B=\cap B(K_i,U_i)\subseteq {\mathrm{End}}_D(V)$. Since $R$ is dense in ${\mathrm{End}}_D(V)$, there is an $f\in R\cap B$. This implies that $f(v_i)=w_i$ for all $i$. ∎

Remarks.

• If $V$ is finite dimensional over $D$, then any dense ring of linear transformations $R={\mathrm{End}}_D(V)$. This can be easily observed by using the second half of the proof above. Take a basis $v_1,\mathrm{\dots },v_n$ of $V$ and any set of $n$ vectors $w_1,\mathrm{\dots },w_n$ in $V$. Let $g$ be the linear transformation that maps $v_i$ to $w_i$. The above proof shows that there is an $f\in R$ such that $f$ agrees with $g$ on the basis elements. But then they must agree on all of $V$ as a result, which is precisely the statement that $g=f\in R$.

• It can be shown that a ring $R$ is a primitive ring iff it is isomorphic to a dense ring of linear transformations of a vector space over a division ring. This is known as the . It is a generalization of the special case of the Wedderburn-Artin Theorem when the ring in question is a simple Artinian ring. In the general case, the finite chain condition is dropped.

Title	dense ring of linear transformations
Canonical name	DenseRingOfLinearTransformations
Date of creation	2013-03-22 15:38:46
Last modified on	2013-03-22 15:38:46
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 16K40
Synonym	dense ring
Related topic	SchursLemma
Defines	Jacobson Density Theorem