Global dimension of the center

Question

Let $R$ be a ring.

Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global dimension)?

Answer 1

Let $R$ be the path algebra over a field $k$ of the quiver $$\begin{array}{ccc} & \alpha & \\ \bullet & \rightleftarrows & \bullet \\ & \beta & \end{array}$$ subject to the relation $$\beta\alpha=0.$$

So $R$ is $5$-dimensional, with basis $\{e_1,e_2,\alpha,\beta,\alpha\beta\}$, where $e_1$ and $e_2$ are idempotents with $e_1+e_2=1$ and $e_1e_2=0=e_2e_1$, and where $e_1\alpha=\alpha=\alpha e_2$ and $e_2\beta=\beta=\beta e_1$.

Then the global dimension of $R$ is $2$, but the centre of $R$, which is spanned by $1$ and $\alpha\beta$, has infinite global dimension.