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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on $p$-bases of rings


Author: Tomoaki Ono
Journal: Proc. Amer. Math. Soc. 128 (2000), 353-360
MSC (1991): Primary 13N05; Secondary 13B05
Published electronically: July 6, 1999
MathSciNet review: 1623048
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Abstract: Let $R\supseteq R'\supseteq R^p$ be a tower of rings of characteristic $p>0$. Suppose that $R$ is a finitely presented $R'$-module. We give necessary and sufficient conditions for the existence of $p$-bases of $R$ over $R'$. Next, let $A$ be a polynomial ring $k[X_1,\dots ,X_n]$ where $k$ is a perfect field of characteristic $p>0$, and let $B$ be a regular noetherian subring of $A$ containing $A^p$ such that $[Q(B)\, :\, Q(A^p)]=p$. Suppose that $Der_{A^p}(B)$ is a free $B$-module. Then, applying the above result to a tower $B\supseteq A^p\supseteq B^p$ of rings, we shall show that a polynomial of minimal degree in $B-A^p$ is a $p$-basis of $B$ over $A^p$.


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Additional Information

Tomoaki Ono
Affiliation: Tokyo Metropolitan College of Aeronautical Engineering 8-52-1, Minami-senju, Arakawa-ku, Tokyo 116-0003, Japan
Email: tono@kouku-k.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05029-7
PII: S 0002-9939(99)05029-7
Keywords: $p$-basis, derivation, Galois extension
Received by editor(s): January 19, 1998
Received by editor(s) in revised form: April 1, 1998
Published electronically: July 6, 1999
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1999 American Mathematical Society