Reflexive primes, localization and primary decomposition in maximal orders

Abstract:

If $R$ is a maximal order and $P$ a reflexive prime ideal of $R$, then the Goldie localization of $R$ at $P$ is shown to be the classical (partial) quotient ring of $R$ with respect to the Ore set $C(P) = \{ r \in R|rx \in P \Rightarrow x \in P\}$. This is accomplished by introducing new symbolic powers of the prime $P$ which agree with Goldie’s symbolic powers. As a consequence, whenever $P$ is a reflexive prime ideal of $R$ and ${P^{(n)}}$ the $n$th (Goldie) symbolic power of $P$, then an ideal $B$ is reflexive if and only if $B = \bigcap \nolimits _{i = 1}^n {P_i^{({n_i})}}$ for uniquely determined reflexive primes ${P_i}$ and integers ${n_i} > 0$. More generally, each bounded essential right (left) ideal is shown to have a reduced primary decomposition and an explicit determination of the components is given in terms of the bound of the ideal.