All rings and algebra in this question are commutative and contains unity.
Suppose $M$ is an $A$ module and $A$ a $R$ algebra. If $pd_R(M) < \infty$, then will that imply $pd_A(M) < \infty$? In particular if $M = \frac{R[X_1, X_2, \ldots,X_n]_N}{I}$, $A = R[X_1, X_2, \ldots,X_n]_N$ where $R[X_1, X_2, \ldots,X_n]$ is polynomial algebra , $N$ a multiplicative closed set and $I$ an ideal, is the answer affirmative?