proof of characterizations of the Jacobson radical
Next, we will prove cyclical containment. Observe that 5) follows after the equivalence of 1) - 4) is established, since 4) is independent of the choice of left or right ideals.
- 1) 2)
Let and take . Let . Then .
Assume is not left invertible; therefore there exists a maximal left ideal of such that .
Note then that . Also, by definition of , we have . Therefore ; this contradiction implies is left invertible.
We claim that 3) satisfies the condition of 4).
We shall first show that is an ideal.
Clearly if , then . If , then
Now there exists such that , hence
Similarly, there exists such that , therefore
Now if , to show that it suffices to show that is left invertible. Suppose , hence , then .
Therefore is an ideal.
Now let . Then there exists such that , hence , so is left invertible.
So there exists such that , hence , then . Thus and therefore is a unit.
Let be the largest ideal such that, for all , is a unit. We claim that .
Suppose this were not true; in this case strictly contains . Consider with and . Now , and since , then for some unit .
So , and clearly since . Hence is also a unit, and thus is a unit.
Thus is a unit for all . But this contradicts the assumption that is the largest such ideal. So we must have .
We must show that if is an ideal such that for all , is a unit, then for every irreducible left -module .
Suppose this is not the case, so there exists such that . Now we know that is the largest ideal inside some maximal left ideal of . Thus we must also have , or else this would contradict the maximality of inside .
But since , then by maximality , hence there exist and such that . Then , so is a unit and . But since is a proper left ideal, this is a contradiction.
|Title||proof of characterizations of the Jacobson radical|
|Date of creation||2013-03-22 12:48:56|
|Last modified on||2013-03-22 12:48:56|
|Last modified by||rspuzio (6075)|