**Note.** To remind the reader again that if and are -algebras and is a field, then in we can identify and with and respectively.

**Theorem**. Let be a central simple -algebra and let be an arbitrary -algebra. If is a two-sided ideal of then

*Proof*. Among all non-zero elements of choose

where and such that is minimal. Then the set is linearly independent over because otherwise one of say would be a linear combination of and that would give us such that contradicting the minimality of Similarly, the set is linearly independent over Now, since is simple and is a two-sided ideal of and it contains the non-zero element we have Thus there exist and a positive integer such that

Let Then by (1). Let

*Claim 1*. Because is a two-sided ideal and Thus

*Claim 2*. To see this, let Then (2) gives us

because But, since we will get by the minimality of Thus, by (3) and Lemma 1, we must have for all That means Hence by (2)

*Claim 3*. Suppose to the contrary that Then, by Lemma 1, and so for all because is -linearly independent and So Contradiction!

The link to the lemma 1 are unavailable. Thanks