Let Λ be an artin algebra and $0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$ a chain of ideals of Λ such that $(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$ for any $0\leq i\leq n-1$ and $\Lambda/I_{n}$ is semisimple. If either none or the direct sum of exactly two consecutive ideals has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. As a consequence, we have that if either none or the direct sum of exactly two consecutive terms in the radical series of Λ has infinite projective dimension, then the finitistic dimension conjecture holds for Λ. Some known results are obtained as corollaries.

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理想的最终维度和链条件

设Λ是一个artin代数并且$0=I_{0}\subseteq I_{1} \subseteq I_{2}\subseteq\cdots \subseteq I_{n}$Λ 的一系列理想使得$(I_{i+1}/I_{i})\rad(\Lambda/I_{i})=0$对于任何$0\leq i\leq n-1$和$\Lambda/I_{n}$是半简单的。如果没有一个或恰好两个连续理想的直接和具有无限的射影维数，则有限维数猜想对 Λ 成立。因此，我们有如果Λ的激进级数中没有一个或恰好两个连续项的直接和具有无限的射影维数，那么有限维数猜想对Λ成立。一些已知的结果作为推论获得。