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# Localization in a Principal Right Ideal Domain

Raymond A. Beauregard
Proceedings of the American Mathematical Society
Vol. 31, No. 1 (Jan., 1972), pp. 21-23
DOI: 10.2307/2038504
Stable URL: http://www.jstor.org/stable/2038504
Page Count: 3
Let $R$ be a principal right ideal domain with right $D$-chain $\{ R^{(\alpha)} \mid 0 \leqq \alpha \leqq \delta \}$, 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 $\delta > \alpha \geqq \lambda$.