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*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.

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

## Comments

## Leave a comment