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In case F = Zp it is clear that there are only finitely many possible mg (X). Hence the result also follows. 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.

Proof. Suppose Rr is a minimal left ideal. To prove rR is a minimal right ideal, it is sufficient to show that if 0 = rr ∈ rR then r ∈ rr R.

Conversely, let I = annR (M ) where M is a simple left R-module. Then, one can consider M as a left R/I-module. The latter is still simple and it is faithful. 5 The Jacobson radical of a ring R is the intersection of all left (respectively right) primitive ideals in R. 5. 1 Let R be a ring. 1. If R is simple, then R is left and right primitive. 2. e. J(R) = {0}) and prime. Proof. To prove part one, assume R is simple and let M be a nonzero left R-module. Because annR (M ) is a two-sided ideal of R we get that annR (M ) = {0}.

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Associative algebra by Eric Jespers

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