Geodesic
On the global dimension of an algebra
Matematičeskie zametki, Tome 14 (1973) no. 3, pp. 399-406 
V. E. Govorov. On the global dimension of an algebra. Matematičeskie zametki, Tome 14 (1973) no. 3, pp. 399-406. http://geodesic.mathdoc.fr/item/MZM_1973_14_3_a10/
@article{MZM_1973_14_3_a10,
     author = {V. E. Govorov},
     title = {On the global dimension of an algebra},
     journal = {Matemati\v{c}eskie zametki},
     pages = {399--406},
     year = {1973},
     volume = {14},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_3_a10/}
}
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%J Matematičeskie zametki
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let algebra $R=\Lambda/P$, where $\operatorname{w. gl. dim}R:=\{\min n|_{\forall R}\text{-modules }X,Y$, $\operatorname{Tor}_{n+1}^R(X,Y)=0\}$. In order that $\operatorname{w. gl. dim}R\le2n$ ($\operatorname{w. gl. dim}R\le2n+1$), it is necessary and sufficient that, for any two ideals of algebra $\Lambda$, a left ideal $A$ and a right ideal $B$, containing ideal $P$, the following equation holds: $$ AP^n\cap P^nB=AP^nB+P^{n+1} \quad (AP^nB\cap P^{n+1}=AP^{n+1}+P^{n+1}B). $$