Heisenberg algebras and coefficient rings*

Abstract

In the noncommutative theory of local rings, the existence of coefficient fields is not always granted. We study a counter-example constructed using an enveloping algebra of a Heisenberg algebra to see how to describe a good coeficient ring for a non-commutative local ring, with commutative residue field. Fhrther, dealing with the case of a noncommutative residue division algebra, we use a theorem of Hochschild on the Brauer group to describe a canonical coefficient ring in the case when the exponent of the residue division algebra is prime to the characteristic of the residue field.*