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 *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 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 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.

nonzero

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