Embedded prime

Remarks

Example: for $I = (x^2, xy)$ in $k[x, y]$ over a field $k$, the associated primes are $(x)$ and $(x, y)$, and $(x) \subsetneq (x, y)$, so $(x, y)$ is embedded. Non-example: $(x)$ is not embedded for the same $I$, because no associated prime is strictly smaller. The minimal primes of $I$ in this sense are exactly the minimal prime ideals over $I$. In a primary decomposition, the components at embedded primes are not unique, while those at minimal primes are. The same terms apply to the associated primes of a module.