For the first part see here.

**6)** If is simple, then

*Proof*. Let Then from we get and thus Hence for every we’ll have which gives us Also, since is simple, which means for some Thus and so Therefore which proves Conversely, let and . Since we have and thus and so

**7)** The left uniform dimension of and are equal.

*Proof*. We saw in the previous section that the left ideals of are exactly in the form where is a left ideal of Clearly is direct iff is direct.

**8) **Let be a nilpotent ideal of and let be the right annihilator of in Then is an essential left ideal of and hence is an essential left ideal of

*Proof*. Let be the right annihilator of in For an essential left ideal of the left ideal of is essential in because for every non-zero left ideal of So we only need to prove the first part of the claim. Let be any non-zero left ideal of and put Then

**9)** If is semisimple, then is semiprime.

*Proof*. So we need to prove that has no non-zero nilpotent ideal. Suppose that is a nilpotent ideal of and let be the right annihilator of in Since is semisimple, for some left ideal of But, from the previous fact, we know that is essential in and thus i.e. Thus for some So and Thus

We proved, in the previous section, that if is prime, then is prime too.

**10)** If is simple, then is prime.

*Proof*. Let be two non-zero ideals of We need to show that We have because and is simple. Therefore for some and We can write for some Then So is a right ideal of which contains a unit. Thus Similarly and hence As a result,