Archive for May, 2012

The Jacobson radical

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even \mathbb{Z} fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one with the property that every element r \in R acts nontrivially in some simple (left) module M.

Said another way, let the Jacobson radical J(R) of a ring consist of all elements of r which act trivially on every simple module. By definition, this is an intersection of kernels of ring homomorphisms, hence a two-sided ideal. A ring R is then semiprimitive if it has trivial Jacobson radical.

The goal of this post will be to discuss some basic properties of the Jacobson radical. I am again working mostly from Lam’s A first course in noncommutative rings.


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