A splitting theorem for algebras over commutative von Neumann regular rings

Abstract:

Let R be a commutative von Neumann ring. Let A be an R-algebra which is finitely generated as an R-module and has $A/N$ separable over R. Here N is the Jacobson radical of A. Then it is shown that there exists an R-separable subalgebra S of A such that $S + N = A$ and $S \cap N = 0$. Further it is shown that if T is another R-separable subalgebra of A for which $T + N = A$ and $T \cap N = 0$, then there exists an element $n \in N$ such that $(1 - n)S{(1 - n)^{ - 1}} = T$. This result is then used to determine the structure of all strong inertial coefficient rings.