1
$\begingroup$

Let $R = L_1 \oplus L_2 \oplus ··· \oplus L_n$ where each $L_i$ is a left ideal and $n \in \mathbb{N}$. Show that there exist idempotents $e_i \in R$ where $e_i = e_i^2$ such that $1_R = \sum\limits_{i=1}^{n} e_i$, $L_i = Re_i$ and $e_i e_j = \delta_{i,j}e_i$ for all i, j.

All I really know that is the elements $\{e_i\}$ are called orthogonal idempotents. I am looking for hints on how to proceed not a premade solution.

How would I also start to prove a variant of this where $e_i \in Z(R)$?

$\endgroup$
  • $\begingroup$ If you have a decomposition into left ideals there is not necessarily a "variant where $e_i\in Z(R)$". Take, for instance, $M_n(F)$ for $F$ a field. You can decompose $1$ into $n$ pairwise orthogonal idempotents, but the ring only has two central idempotents. But if you have decomposed the ring into a finite "direct sum of ideals" then yes, you can find central idempotents. $\endgroup$ – rschwieb Aug 5 '18 at 2:22
1
$\begingroup$

Start by decomposing $1_R$ as a sum $\sum_{i=1}^ne_i$, where $e_i\in L_i$. This decomposition is unique, so you can actually be sure that these $e_i$'s must be the ones you want. Then, proceed to showing that they satisfy the conditions. You may deduce relations between the $e_i$'s by using the fact that $1_R$ is idempotent. Can you conclude with this?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.