The main result of this paper is that if \(R\) is any semiprime ring with left Krull dimension \(\alpha\), then the left global homological dimension of \(R\) equals the supremum of the projective dimensions of the \(\beta\)- critical cyclic left \(R\)-modules for \(\beta < \alpha\). The same conclusion is also obtained for full upper triangular matrix rings over domains, and for certain formal triangular matrix rings; further, analogs for rings with Gabriel dimension are given. These results continue a line developed by M. L. Teply, who proved the analogous formula for the weak global dimension of any right coherent ring with left Krull dimension [Bull. Aust. Math. Soc. 39, 215-223 (1989; Zbl 0654.16018)], motivated by work on the problem of whether the global dimension of any noetherian ring equals the supremum of the projective dimensions of its simple modules. A discussion of this problem can be found in the book by the reviewer and R. B. Warfield [An introduction to noncommutative Noetherian rings (Cambridge Univ. Press, 1989; Zbl 0679.16001), p. 287].