proof of characterizations of the Jacobson radical
First, note that by definition a left primitive ideal is the annihilator of an irreducible left -module, so clearly characterization
1) is equivalent
to the definition 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)
We know that every left primitive ideal is thelargest ideal contained in a maximal left ideal. So the intersection
of all left primitive ideals will be contained in the intersection of allmaximal left ideals.
- 2) 3)
Let and take .Let . Then .
Assume is not left invertible; therefore there exists a maximalleft ideal of such that .
Note then that . Also, by definition of , we have. Therefore ; this contradiction
implies isleft invertible.
- 3) 4)
We claim that 3) satisfies the condition of 4).
Let .
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
Hence .
Now if , to show that it suffices toshow that is left invertible. Suppose , hence, then .
So .
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 aunit. We claim that .
Suppose this were not true; in this case strictly contains .Consider with and .Now , and since , then for someunit .
So , and clearly since. Hence is also a unit, and thus isa unit.
Thus is a unit for all . But this contradicts theassumption
that is the largest such ideal. So we must have.
- 4) 1)
We must show that if is an ideal such thatfor 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 maximalleft ideal of . Thus we must also have ,or else this would contradict the maximality of inside .
But since , then by maximality , hence thereexist and such that . Then , so is a unit and . But since is a proper left ideal, thisis a contradiction.