Existence of maximal orders
The author calls a commutative ring O pseudo-Noetherian if for any \(a\in O\) the set of prime ideals containing a, P(a), has only a finite number of minimal elements and for any minimal \(p\in P(a)\) the ring \(O_ p\) is Noetherian. Let \(K\) be the quotient field and P the set of prime ideals of height 1 of the integral pseudo-Noetherian ring O. The following criterion for the existence of maximal O-orders is proved: A maximal O- order exists in a finite-dimensional K-algebra A if and only if the following two conditions are satisfied: (1) for any \(p\in P\), the algebra \(\bar A_ p=A\otimes_ O\bar O_ p\), where \(\bar O_ p\) is the p-adic completion of \(O_ p\), is semisimple; (2) for some (and then also for any) O-order \(\Lambda\) in the algebra A and all \(p\in P\), except for a finite number, the \(O_ p\)-orders \(\Lambda_ p\) are maximal.
scientific article; zbMATH DE number 3933249