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On the coherence and weak dimension of the rings $ R\langle x\rangle$ and $ R(x)$


Author: Sarah Glaz
Journal: Proc. Amer. Math. Soc. 106 (1989), 579-587
MSC: Primary 13F20
DOI: https://doi.org/10.1090/S0002-9939-1989-0961405-9
MathSciNet review: 961405
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Abstract: Let $ R$ be a commutative ring. We first derive necessary and sufficient conditions for the rings $ R\left\langle x \right\rangle $ and $ R(x)$ to be coherent. Next, for stably coherent rings of finite weak dimension exact relations are found between the weak dimension of $ R$ and that of $ R\left\langle x \right\rangle $ and $ R(x)$. These relations are used to determine necessary and sufficient conditions for $ R\left\langle x \right\rangle $ and $ R(x)$ to be Von Neumann regular or semihereditary.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0961405-9
Keywords: Regular rings, coherence, Von Neumann regular, semihereditary, depth, grade, regular sequence
Article copyright: © Copyright 1989 American Mathematical Society