Suppose $R$ is a semisimple ring and if $L$ is a minimal left ideal. Let $B$ be the direct sum of all minimal left ideals isomorphic to $L$ ($B$ is called a simple component corresponding to $L$). It is a standard result that $B$ is a two-sided ideal. The argument critically uses the fact that $R$ is semisimple.


I am searching for an example of a ring $R$ and a simple component that is not a two-sided ideal.


Since, we need a minimal left ideal, we may consider the ring to be Artinian as they have always have a minimal left ideal.
I first tried considering the endomorphism ring of a vector space with a countable basis, but later I learned that it is neither semisimple nor Artinian.


I figured out that we cannot consider a simple Artinian ring, for it is semisimple. Also, we cannot consider any Artinian ring $R$ whose Jacobson radical $J(R)=0$ for the same reason.


Does anyone know any example?



Simple component that is not a two-sided ideal
I'm not sure why you believe it depends on $R$ being semisimple. The usual argument goes like this and makes no reference to semisimplicity:


If $r\in R$ and $L'\cong L$ where $L'$ is a left ideal of $R$, then by right multiplication, $r$ defines a homomorphism of left $R$ modules $L'\to R$. Because $L'$ is simple, either $L'r=\{0\}$ or $L'r\cong L'$. Either way, this shows right multiplication sends simple left ideals isomorphic to $L$ into the simple component for $L$, i.e. the component is an ideal of $R$.


Now, one thing that might not happen in general is for the component to be a summand. For example, in the full linear ring of transformations of a countable dimensional vector space, there is certainly a nonzero component there (in fact it is the only nontrivial ideal), but the ring is also prime, and so it cannot be decomposed into pieces.


Certainly the name "simple component" is meant to reflect the fact that in semisimple Artinian rings, the simple components are actually simple rings in their own right, and their product is the semisimple Artinian ring. But the existence of the ideal itself is not confined to semisimple rings.


