Is it possible ?
1 Answer
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3
Rather obviously yes.
Let $A$ be the algebra over the field $K$ generated by elements $a_1,a_2,\ldots,$ with $a_i$ in dimension $i$ and with $a_ia_j=0$ for all $i$ and $j$. This is an incredibly uninteresting example, but since each graded piece is onedimensional, its Poincare series is $\sum_{n=0}^\infty t^n=1/(1t)$.

$\begingroup$ I agree but I am interested with an algebra with nontrivial multiplication. $\endgroup$– MelaniaSep 29, 2010 at 20:47

3$\begingroup$ @Melania: how about start as Robin did, and then kill all $a_ia_ja_k$? $\endgroup$ Sep 29, 2010 at 21:23

$\begingroup$ Yes..The Poincare series will be $\dfrac{1}{(1t)(1t^2)}.$ Thanks! $\endgroup$– MelaniaSep 29, 2010 at 21:54