The Weak (Gorenstein) Global Dimension of Coherent Rings with Finite Small Finitistic Projective Dimension

The small finitistic dimension of a ring is determined as the supremum projective dimensions among modules with finite projective resolutions. This paper seeks to establish that, for a coherent ring <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>R</a:mi></a:math> with a finite weak (resp. Gorenstein) global dimension, the small finitistic dimension of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>R</c:mi></c:math> is equal to its weak (resp. Gorenstein) global dimension. Consequently, we conclude some new characterizations for (Gorenstein) von Neumann and semihereditary rings.