Let a ring $R$ has finite Goldie dimension, say, as left $R$-module. Could it be deduced that $R$ is $I$-finite in the sense that $R$ does not contain an infinite subset of non-zero orthogonal idempotents?
Referring to T.Y.Lam's "Lectures on Modules and Rings", each direct summand $C$ of a module $M$ is a complement in $M$; in fact, when $M=C\oplus S$ then $C$ is a complement to $S$. On the other hand, by Proposition (6.30) of the same book, infinite-ness of Goldie dimension for a module $M$ is equivalent to existence of an infinite strictly ascending chain of complements in $M$.
Now, my try is to assume an infinite subset $\{e_{\alpha }\}$ of orthogonal idempotents in the ring $R$ and to consider the infinite direct sum $\bigoplus_ {\alpha} Re_{\alpha }$. Is this a valid argument?
Thanks for any help!
Yes, in fact this already appears in that book.
On page 229 you can see "$DCC_{III}\implies DCC_V$ which, after reading Prop 6.30 ($DCC_{III}=$ finite uniform dimension) and Prop 6.59 ($DCC_{V}=$ orthogonally finite) is what you are asking.