I am reading Rings and Categories of Modules by Frank W.Anderson. In the book on page 161:
$End(M_{D})$ is primitive for every vector space $M_{D}$, but if $M_{D}$ is infinite dimensional, then $End(M_{D})$is not simple.
I want to know why $End(M_{D})$ is not simple when $M_{D}$ is infinite dimensional.
Because the ideal consisting of transformations with finite dimensional images is a proper ideal.
Actually, if $\kappa$ is the cardinality of the basis of $M_D$, for each infinite cardinal $\gamma$ less than $\kappa$, there is such an ideal of elements having dimensionality strictly less than $\gamma$, and those turn out to be all the ideals. Consequently the ideals are linearly ordered and have this nice description.