Krull and global dimensions of semiprime Noetherian PI-rings

Resco, Richard; Small, Lance W.; Stafford, J. T.

doi:10.1090/S0002-9947-1982-0670932-1

Krull and global dimensions of semiprime Noetherian PI-rings
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by Richard Resco, Lance W. Small and J. T. Stafford

Trans. Amer. Math. Soc. 274 (1982), 285-295

DOI: https://doi.org/10.1090/S0002-9947-1982-0670932-1

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Abstract:

In this paper it is shown that if $R$ is a semiprime Noetherian PI-ring of finite global dimension, then the Krull dimension of $R$ is less than or equal to its global dimension. The proof depends upon two preliminary results on arbitrary Noetherian PI-rings, which are of independent interest: (i) any height two prime ideal of $R$ contains infinitely many height one prime ideals; (ii) the localization of the polynomial ring $R[x]$ at its set of monic elements is a Jacobson ring.

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