From the introduction: In [

*W. D. Burgess*, Proc. Edinb. Math. Soc., II. Ser. 30, 351-362 (1987;

Zbl 0608.16028)] and [

*K. R. Fuller*, Contemp. Math. 124, 51-72 (1992;

Zbl 0746.16013)] we employed an equivalence relation on the indecomposable projective modules over a left Artinian ring

$R$ that induces the “finest” presentation of

$R$ as a ring of lower block triangular matrices. The equivalence classes correspond to the simply connected components of the quiver of

$R$ and they contain information on such things as the Cartan determinant and the finitistic dimensions of

$R$. Our main purpose here is to show that such presentations provide bounds on the finitistic dimensions of certain rings and Artin algebras that yield generalizations of some results of

*M. I. Platzeck* and

*F. U. Coelho* [in Bol. Soc. Mat. Mex., III. Ser. 7, No. 1, 49-57 (2001;

Zbl 1011.16011) and Commun. Algebra 24, No. 8, 2515-2533 (1996;

Zbl 0857.16012)]. In the process, we provide a simple proof of an old result from [

*R. M. Fossum, P. A. Griffith, I. Reiten*, Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory. Lect. Notes Math. 456 (1975;

Zbl 0303.18006)] and point out that a theorem of

*S. O. Smalø* [Proc. Am. Math. Soc. 111, No. 3, 651-656 (1991;

Zbl 0724.16003)] provides a connection between the functorial finiteness in

$R$-mod of the category

${\mathcal{P}}^{<\infty}\left(R\right)$ of modules of finite projective dimension, and the analogous categories over the irreducible components of

$R$.