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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Finite generation of Noetherian graded rings


Authors: Shiro Goto and Kikumichi Yamagishi
Journal: Proc. Amer. Math. Soc. 89 (1983), 41-44
MSC: Primary 13E05; Secondary 13E15
DOI: https://doi.org/10.1090/S0002-9939-1983-0706507-1
MathSciNet review: 706507
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Abstract: Let $ H$ be an additive abelian group. Then a commutative ring $ A$ is said to be $ H$-graded if there is given a family $ {\{ {A_h}\} _{h \in H}}$ of subgroups of $ A$ such that $ A = { \oplus _{h \in H}}{A_h}$ and $ {A_h}{A_g} \subset {A_{h + g}}$ for all $ h$, $ g \in H$. In this note it is proved, provided $ H$ is finitely generated, that an $ H$-graded ring $ A$ is Noetherian if and only if the ring $ {A_0}$ is Noetherian and the $ {A_0}$-algebra $ A$ is finitely generated. This is not true in general unless $ H$ is finitely generated. A counterexample is given.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0706507-1
Keywords: $ H$-graded rings
Article copyright: © Copyright 1983 American Mathematical Society