Embedded primary component
= Embedded primary component
A component in a <minimal primary decomposition> is embedded if its <radical of an ideal> strictly contains another radical occurring in that decomposition. Such components need not be unique. For example,
$$
(x^2,xy)=(x)\cap(x^2,y)=(x)\cap(x^2,y-x)
$$
in $k[x,y]$ gives different $(x,y)$-<primary ideals> as embedded components, with the same isolated component $(x)$.