# Answer to Question #23263 in Abstract Algebra for Melvin Henriksen

Question #23263

Let R = End(Vk) where V is a vector space over a division ring k. Show that all ideals of R are linearly ordered by inclusion and idempotent.

Expert's answer

If dim

Therefore, it suffices to treat thecase when

The ideals of

To show that they are allidempotent, consider any ideal nonzero

There exists an infinite cardinal

For any

*k V <∞*,*R*isa 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 allidempotent, consider any ideal nonzero

*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 theidentity 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 wehave proved that*I*=*I^*2.
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