# Answer to Question #23262 in Abstract Algebra for Melvin Henriksen

Question #23262

Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals not equal R are prime.

Expert's answer

It is sufficient to show that allideals of

nonzero

*R*are linearly ordered by inclusion and idempotent. If dim*kV <∞*,*R*is a simple ring. Therefore, it suffices to treat thecase when*V*is infinite-dimensional. The ideals of*R*are linearlyordered by inclusion. To show that they are all idempotent, consider any idealnonzero

*I*. There exists an infinite cardinal*β**<*dim*k V*such that*I*=*{f**∈**R*: dim*k f*(*V*)*<**β**}.*For any*f**∈**I*, let*f'**∈**R*be such that*f'*is the identity on the*f*(*V*), and zero on a direct complement of*f*(*V*). Clearly,*f'**∈**I*, and*f*=*f'f*. Therefore,*f**∈**I^*2, and we have proved that*I*=*I^*2.
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