Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be an ideal. A monomial quotient $S/I$ is called Cohen-Macaulay if the projective dimension of $S/I$ is equal to the smallest number of generators of any irreducible component of $I$.

Is there a name for the "opposite" property that the projective dimension of $S/I$ is equal to the largest number of generators of any irreducible component of $I$? Are there prior work on these rings?

Let $S = \mathbb k[x_1, \ldots, x_n]$ be a polynomial ring over a field $\mathbb k$ and let $I \subseteq S$ be a monomial ideal. For a monomial ideal $J$, let $\#(J)$ be the smallest number of monomials that generate $J$. A monomial quotient $S/I$ is called Cohen-Macaulay if the projective dimension of $S/I$ is equal to $\min\{\#(J): J \text{ is an irreducible component of }I\}$.

Is there a name for the "opposite" property that the projective dimension of $S/I$ is equal to $\max\{\#(J): J \text{ is an irreducible component of }I\}$? Are there prior work on these rings?