Global dimension of ring \(T=\left(\begin{smallmatrix} R &M\\ N &S\end{smallmatrix}\right)_{(\theta,\psi)}\) (Q1906581)

Let \(\text{LGD}(-)\) denote the left global dimension of a ring. All rings in the paper are left Noetherian. Let \(T=(\begin{smallmatrix} R\\ N\end{smallmatrix} \begin{smallmatrix} M\\ S\end{smallmatrix})_{(\theta,\psi)}\), where \(R\) and \(S\) are rings, \(_RM_S\) and \(_SN_R\) are bimodules, and \(\theta:M\otimes_S N\to R\) and \(\psi:N\otimes_R M\to S\) are bimodule homomorphisms. The authors prove that if \(T\) is left Noetherian and \(M_S\) is flat then \[ \max\{\text{LGD}(R), \text{LGD}(S)\}\leq\text{LGD}(T)\leq 1+\max\{\text{LGD}(S), 1+\text{pd}(_SN)+\text{LGD}(R/MN)\}. \]