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For a quantum torus $C_q[x_1^{\pm1}, ...,x_n^{\pm1}]$ satisfying $x_ix_j=q_{ij}x_jx_i$.

Question: Is this quantum torus a principal ideal domain?

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victor
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  • You had better edit it with LaTeX. – eccstartup Jul 12 '13 at 05:59
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    At the level of generality at which you posed the question the answer is clearly no: take all $q_{ij}$ equal to $1$, so that you have a commutative ring of Laurent polynomials in $n$ variables. The Krull dimension of such a thing is $n$, which is usually not $1$, which is the Krull dimension of a principal ideal domain which is not a field. – Mariano Suárez-Álvarez Jul 12 '13 at 06:37
  • I like your question, rather to ask further under which condition sit might be a principal ideal domain? – al-Hwarizmi Jul 12 '13 at 06:42
  • What is a noncommutative principal ideal domain? – Qiaochu Yuan Jul 12 '13 at 07:29

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