Localization in a Principal Right Ideal Domain

Abstract

Let $R$ be a principal right ideal domain with right $D$-chain $\{ {R^{(\alpha )}}|0 \leqq \alpha \leqq \delta \}$, and let ${K_\alpha } = R{({R^{(\alpha )}})^{ - 1}}$ be the associated chain of quotient rings of $R$. The local skew degree of $R$ is defined to be the least ordinal $\lambda$ such that ${K_\lambda }$ is a local ring. The main result states that for each $\alpha \geqq \lambda ,{K_\alpha }$ is a local ring; equivalently, $R$ has a unique $(\alpha + 1)$-prime for $\delta > \alpha \geqq \lambda$.