Associative algebra by Eric Jespers

By Eric Jespers

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It follows that aR is nilpotent. This proves the result. 11 If R is a left Noetherian ring, then the prime radical contains all nil left and right ideals. Thus, B(R) = Nil(R) = N1 (R). 1 (Brauer) Let L be a minimal left ideal in a ring R. Then either L2 = {0} or L = Re for some idempotent e ∈ L. Proof. Suppose L2 = {0}. Then La = {0} for some a ∈ L and thus La = L. Let e ∈ L so that a = ea. annL (a) = {l ∈ L | la = 0}. Clearly A is a left ideal of R and A ⊂ L. Hence A = {0} . Because e2 − e ∈ L and (e2 − e)a = e(1 − e)a = e(a − ea) = 0 we get e2 − e ∈ A = {0}.

6 (Schur) Let k be a field. A finitely generated periodic subgroup G of GLn (k) is finite. Proof. 5 G has finite exponent. 2 (only in the last part does the proof differ in case k n is not a simple kG-module). Consider again the mapping ϕ : G → G1 × G2 . By induction the groups G1 and G2 are finite. Hence ker ϕ = I h 0 I ∈G is of finite index in G. 4 the implies that G is finite. ✷ A group G is said to be locally finite if every finitely generated subgroup is finite. 7 A linear group G ⊆ GLn (K) over a field K is periodic if and only if G is locally finite.

6. 6 39 Linear groups and the Burnside problem In Algebra I we showed that there is a close relationship between the representation theory of a finite group G and the module and ring structure of the finite dimensional algebra kG. We proved this for an algebraically closed field k and in case char(k) does not divide the order of the group G. One can also show this for some fields that are not necessarily algebraically closed. We now show that in the study of infinite groups G one can sometimes also make use of ring and module theory.

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