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Coherence for polynomial rings
Journal article   Open access   Peer reviewed

Coherence for polynomial rings

Victor Camillo
Journal of algebra, Vol.132(1), pp.72-76
1990
DOI: 10.1016/0021-8693(90)90252-J
url
https://doi.org/10.1016/0021-8693(90)90252-JView
Published (Version of record) Open Access

Abstract

Let II= IIR, be an arbitrary product of copies of R,. We say R is right n-coherent if every finitely generated submodule of Z7 is finitely presented. This notion is called strong coherence in [7], where it seems to have been invented. Here we obtain a new characterization of n-coherence, and study Z7coherence for polynomial rings. Using Goldie’s Theorem, we show that if R is a semiprime two-sided noetherian ring then R[S] is right n-coherent for any set of variables S (Theorem 6). We also obtain the same result for two-sided noetherian rings R, not assumed to be semiprime, provided R contains an uncountable field. Call ring R a left *-ring (star ring) if it has the property that HomA , RR) = ( )* takes finitely generated left R-modules to finitely generated right R modules. The above ideas are connected by the following:

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