THE FINITISTIC DIMENSION AND CHAIN CONDITIONS ON IDEALS

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.