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Chains of Prime Ideals and Primitivity of \(\mathbb {Z}\)-Graded Algebras

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Abstract

In this paper we provide some results regarding affine, prime, \(\mathbb {Z}\)-graded algebras \(R=\bigoplus _{i\in \mathbb {Z}}R_{i}\) generated by elements with degrees 1,−1 and 0, with R 0 finite-dimensional. The results are as follows. These algebras have a classical Krull dimension when they have quadratic growth. If R k ≠0 for almost all k then R is semiprimitive. If in addition R has GK dimension less than 3 then R is either primitive or PI. The tensor product of an arbitrary Brown-McCoy radical algebra of Gelfand Kirillov dimension less than three and any other algebra is Brown-McCoy radical.

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Correspondence to André Leroy.

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Presented by Paul Smith.

A. Smoktunowicz was supported by ERC Advanced grant Coimbra 320974 and M. Ziembowski was supported by the Polish National Science Centre grant UMO-2013/09/D/ST1/03669.

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Greenfeld, B., Leroy, A., Smoktunowicz, A. et al. Chains of Prime Ideals and Primitivity of \(\mathbb {Z}\)-Graded Algebras. Algebr Represent Theor 18, 777–800 (2015). https://doi.org/10.1007/s10468-015-9516-0

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  • DOI: https://doi.org/10.1007/s10468-015-9516-0

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