If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R)i = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B)n, the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following.
If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms gi,...,gk of B/N(B) over F such that B is a homomorphic image of B/N[[x1,…,xk;g1,…,gk]] the power series ring over B/N(B) in noncommuting indeterminates xi, where xib = gi(b)xi for all b ϵ B/N.
Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms g1,…,gk of a v-ring V such that B is a homomorphic image of V [[x1,…,xk;g1,…,gk]].
In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B.
