The Dynkin and the extended Dynkin graphs are characterized by representations over the real numbers.

We generalize the relative (co)tilting theory of Auslander-Solberg in the category mod Λ of finitely generated left modules over an artin algebra Λ to certain subcategories of mod Λ. We then use the theory (relative (co)tilting theory in subcategories) to generalize one of the main result of Marcos et al. [Comm. Algebra 33 (2005)].

Let $R$ be a ring, $n$ a fixed non-negative integer, $𝒲ℐ$ the class of all left $R$-modules with weak injective dimension at most $n$, and $𝒲ℱ$ the class of all right $R$-modules with weak flat dimension at most $n$. Using left (right) $𝒲ℐ$-resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that $-\otimes -$ is right balanced on ${ℳ}_R\times {}_Rℳ$ by $𝒲ℱ\times 𝒲ℐ$, and investigate the global right $𝒲ℐ$-dimension of ${}_Rℳ$ by right derived functors of $\otimes$.