# Algebra [Lecture notes] by I. M. Isaacs

By I. M. Isaacs

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To see this, we suppose that it is not true, and let Y = {V ⊆ M |N < V }. Our hypothesis ensures that M ∈ Y, so we know that Y is not empty. By the minimal condition, we see that Y has a minimal element, say K. By the minimality of K, K/N is X-simple. Yet then K > N and K has an X-composition series, which contradicts the maximality of N . Therefore Y must be empty, and we have that N = M . The preceding proof is an excellent example of how one uses the maximal/minimal condition. It is quite canonical, in that sense.

The above proof is a very standard application of Zorn’s lemma and is quite a good canonical example of how to apply it. We previously introduced the notion of the annialator of an element, ann(m). We additionally introduced ann(M ) = {r ∈ R|M r = 0}. Another convenient way to view ann(M ) is as the intersection of all of the annialators of the individual elements; that is, ann(M ) = {ann(m)|m ∈ M }. It is a convenient fact that ann(M ) is an ideal of R, not just a right ideal. 2 The Jacobson Radical The final concept we present for the semester is that of the Jacobson Radical.

By the “directness” of the product, we have that sk = tk for all k, so g = ti = si = 1. 34 We draw as a corollary the following important fact. 1. If G is the internal direct product of Ni for 1 ≤ i ≤ r then Ni ∩ Nj = 1 for i = j and so the elements of Ni commute with the elements of Nj . We now proceed with the proof of the theorem. We wish to exhibit an isomorphism θ from the external direct product of the Ni to G. Let θ(n1 , n2 , . . , nr ) = n1 n2 . . nr , which is in G. Note that θ is surjective as G is the internal direct product of the Ni .