[Math] Jacobson radical = intersection of all maximal two-sided ideals

noncommutative-algebrara.rings-and-algebrasrt.representation-theory

I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can do is ask it here…

Let $A$ be a finite-dimensional $k$-algebra, where $k$ is a field. Is it true that the Jacobson radical equals the intersection of all maximal two-sided ideals? (The latter intersection is known as the Brown-McCoy radical of $A$.)

If yes, a short proof (the more self-contained, the better) would be great.

(This is again for use in coalgebra theory.)

Best Answer

Yes, this is true; it's essentially just a restatement of Artin-Wedderburn. All you need to do is note that by Artin-Wedderburn, a finite dimensional algebra with trivial Jacobson radical is a sum of matrix algebras over division rings (where it's obvious that the intersection of all maximal ideals is trivial); for an arbitrary ring, kill the Jacobson radical, and apply the result to see you've killed the intersection of maximal ideals.

EDIT: Kevin makes a good point, which is that there are basically two parts of Artin-Wedderburn:

  1. Showing that a semi-simple Artinian ring (in the sense of trivial Jacobson radical) is a direct sum of simple rings (in the sense of no proper two-sided ideals).
  2. Showing that every simple Artinian ring is a matrix ring over a division ring.

You only need 1. for this fact. On the other hand, if I wanted to use this fact in a paper, I would say something like "the Jacobson radical of a finite-dimensional $k$-algebra is the intersection of its maximal two-sided ideals; this follows from Artin-Wedderburn."

Of course, you could cite this MO page.

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