Example 2

A Macaulay ring has no embedded associated prime ideals (No. 2, Prop. 2). Conversely, let $A$ be a noetherian ring of dimension $\leqslant 1$ which has no embedded associated prime ideals; for every nonempty finite subset strongly secant S of A, the A-module $A/SA$ is of dimension $\leqslant 0$, hence Macaulay (No. 1, Example 1); hence A is a Macaulay ring (No. 4, Theorem 2). In particular a reduced noetherian ring of dimension $\leqslant 1$ is a Macaulay ring.