Chapter II. Wedderburn-Artin Ring Theory

Abstract

This chapter studies finite-dimensional associative division algebras, as well as other finite-dimensional associative algebras and closely related rings. The chapter is in two parts that overlap slightly in Section 6. The first part gives the structure theory of the rings in question, and the second part aims at understanding limitations imposed by the structure of a division ring. Section 1 briefly summarizes the structure theory for finite-dimensional (nonassociative) Lie algebras that was the primary historical motivation for structure theory in the associative case. All the algebras in this chapter except those explicitly called Lie algebras are understood to be associative. Section 2 introduces left semisimple rings, defined as rings $R$ with identity such that the left $R$ module $R$ is semisimple. Wedderburn's Theorem says that such a ring is the finite product of full matrix rings over division rings. The number of factors, the size of each matrix ring, and the isomorphism class of each division ring are uniquely determined. It follows that left semisimple and right semisimple are the same. If the ring is a finite-dimensional algebra over a field $F$, then the various division rings are finite-dimensional division algebras over $F$. The factors of semisimple rings are simple, i.e., are nonzero and have no nontrivial two-sided ideals, but an example is given to show that a simple ring need not be semisimple. Every finite-dimensional simple algebra is semisimple. Section 3 introduces chain conditions into the discussion as a useful generalization of finite dimensionality. A ring $R$ with identity is left Artinian if the left ideals of the ring satisfy the descending chain condition. Artin's Theorem for simple rings is that left Artinian is equivalent to semisimplicity, hence to the condition that the given ring be a full matrix ring over a division ring. Sections 4–6 concern what happens when the assumption of semisimplicity is dropped but some finiteness condition is maintained. Section 4 introduces the Wedderburn–Artin radical $\mathrm{rad} R$ of a left Artinian ring $R$ as the sum of all nilpotent left ideals. The radical is a two-sided nilpotent ideal. It is 0 if and only if the ring is semisimple. More generally $R/\mathrm{rad} R$ is always semisimple if $R$ is left Artinian. Sections 5–6 state and prove Wedderburn's Main Theorem–-that a finite-dimensional algebra $R$ with identity over a field $F$ of characteristic 0 has a semisimple subalgebra $S$ such that $R$ is isomorphic as a vector space to $S\oplus\mathrm{rad} R$. The semisimple algebra $S$ is isomorphic to $R/\mathrm{rad} R$. Section 5 gives the hard part of the proof, which handles the special case that $R/\mathrm{rad} R$ is isomorphic to a product of full matrix algebras over $F$. The remainder of the proof, which appears in Section 6, follows relatively quickly from the special case in Section 5 and an investigation of circumstances under which the tensor product over $F$ of two semisimple algebras is semisimple. Such a tensor product is not always semisimple, but it is semisimple in characteristic 0. The results about tensor products in Section 6, but with other hypotheses in place of the condition of characteristic 0, play a role in the remainder of the chapter, which is aimed at identifying certain division rings. Sections 7–8 provide general tools. Section 7 begins with further results about tensor products. Then the Skolem–Noether Theorem gives a relationship between any two homomorphisms of a simple subalgebra into a simple algebra whose center coincides with the underlying field of scalars. Section 8 proves the Double Centralizer Theorem, which says for this situation that the centralizer of the simple subalgebra in the whole algebra is simple and that the product of the dimensions of the subalgebra and the centralizer is the dimension of the whole algebra. Sections 9–10 apply the results of Sections 6–8 to obtain two celebrated theorems—Wedderburn's Theorem about finite division rings and Frobenius's Theorem classifying the finite-dimensional associative division algebras over the reals.