The main result of the paper is motivated by a result of \textit{E. Kirkman} and \textit{J. Kuzmanovich} for non-commutative rings [Pac. J. Math. 134, No. 1, 121-132 (1988; Zbl 0617.16014)] and states that if in a pullback diagram of commutative rings \[ \begin{matrix} A & @>i_ 1>> & A_ 1 \\ \downarrow && \downarrow \\ A_ 2 & \longrightarrow & A_ 0 \end{matrix} \] a map \(i_ 1\) is surjective and \(\text{pd}_{A_ i} (\text{Tor}^ A_ j(A_ i, A/a)) \leq n - j\) \((0 \leq j \leq n,\;i = 1, 2)\) for all ideals \(a\) of \(A\) then \(\text{gl.dim} A \leq n\). A similar results holds for weak global dimension. Interesting examples are computed.