Isotypic components are just simple two-sided ideals

I'm trying to show that when we decompose a semisimple ring $R$ into isotypic components $$R \overset{_R\mathsf{Mod}}{\cong}\bigoplus_{j=1}^{k_1}{I^{(1)}_j} \bigoplus \dotsb \bigoplus \left( \bigoplus_{j=1}^{k_n}I_j^{(n)} \right) \\ = R^{(1)} \oplus \dotsb \oplus R^{(n)}$$ we have actually decomposed it into the simple two-sided ideals. (I.e. the simple two-sided ideals of $R$ are precisely the sums of all simple left ideals of the same isomorphism type.)

I understand why the isotypic components are two-sided ideals. To show that any two-sided ideal is of this form, I was looking on this page, and their explanation is

...characteristic left ideals are precisely two-sided ideals. Thus two-sided ideals are sums of isotypic components.

I don't understand what this means, nor do I know what a "characteristic ideal" is (is it an ideal $I$ such that any $R$ left-linear automorphism acts trivially on $I$?). Can someone help, or just point me to another proof of what I'm trying to prove?

• I think I finally got it, so I posted my own answer. – Eric Auld Mar 15 '15 at 23:53

Remember that given $$\varphi \in \text{End}_{_R\textsf{Mod}}(R)$$, there is some $$a \in R$$ such that $$\varphi$$ is just right multiplication by $$a$$, because $$\varphi(r) = \varphi(r \cdot 1)= r \cdot \varphi(1)$$.

If $$M \in {_R\textsf{Mod}}$$, define $$N$$ to be a characteristic submodule if it is preserved as a set by every (left) endomorphism of $$M$$. A characteristic left ideal is defined the same.

Proposition 1: Let $$I \subset R$$ be a left ideal. Then $$I$$ is two-sided iff $$I$$ is characteristic left ideal of $$R$$.

Proof: $$\text{End}_{_R\textsf{Mod}}(R)$$ is exactly those maps consisting of right multiplication by some element of $$R$$. $$\square$$

Proposition 2: Suppose $$M$$ is a left module over a semisimple ring. Then $$N < M$$ is characteristic iff $$N$$ is a sum of entire isomorphism classes of simple submodules of $$M$$.

Proof: Write $$M = N \oplus N'$$ by semisimplicity, and split $$N$$ and $$N'$$ into finitely many simple submodules. $$N$$ is a sum of entire isomorphism classes of simple submodules of $$M$$ iff for all $$i$$ the canonical projection onto the $$i$$th isotypic component (AKA multiplication by $$R^{(i)}$$) is zero either on $$N$$ or on $$N'$$. Suppose that $$S_1, S_2$$ are isomorphic simple submodules such that $$S_i$$ appears in the sum for $$N$$ and $$S_2$$ appears in the sum for $$N'$$. Then there is an endomorphism which swaps them, and so does not preserve $$N$$ as a set.

Conversely, it is clear that an isotypic component is characteristic, since a map between simple submodules must either be zero or an isomorphism. $$\,\,\square$$

Corollary: If $$R$$ is semisimple, $$I$$ is a simple two-sided ideal iff $$I$$ is an isotypic component of left ideals.