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I'm reading through Lang's algebra. Lang gives the Jacobson density theorem in the following way:

Let $R$ be a ring (with unity) and $E$ a semisimple $R$-module. Let $R' = \operatorname{End}_R(E), \ R'' = \operatorname{End}_{R'}(E)$.

Let $x_1, \ldots, x_n \in E, f \in R''$. Then there exists $r \in R$ such that $$f(x_i) = r.x_i, i = 1, \ldots, n .$$

In other words, $f$ acts as some $r \in R$ on every finitely-generated submodule of $E$. Of course, if $E$ is finitely generated, every element of $R''$ acts as some element of $R$ on $E$.

In every subsequent application of the theorem, $E$ is finitely-generated over $R$. I would like to see some standard/natural examples of situations where $E$ isn't finitely generated and not every element of $R''$ is representable as an element of $R$.

ante.ceperic
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1 Answers1

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Let $R=\mathbb{Z}$ (though this example generalizes to any commutative ring with infinitely many maximal ideals) and $$E=\bigoplus_p\mathbb{Z}/p\mathbb{Z},$$ where the sum is over all primes.

Then $$R'\cong R''\cong \prod_p\mathbb{Z}/p\mathbb{Z},$$ acting on $E$ in the obvious way.

Let $f=(f_p)\in R''$.

Any finite subset $\{x_1,\dots,x_n\}\subset E$ is contained in $\bigoplus_{p\in I}\mathbb{Z}/p\mathbb{Z}$ for some finite set $I$ of primes and by the Chinese Remainder Theorem there is some $r\in\mathbb{Z}$ with $f_p=r\pmod{p}$ for all $p\in I$.

However, for general $f$ there is no $r\in\mathbb{Z}$ with $f_p=r\pmod{p}$ for all primes $p$.

Jeremy Rickard
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