Primary Decomposition

Question

The standard primary decomposition theorem in algebra is about being able to write an ideal uniquely as an intersection of primary ideals. In linear algebra the theorem is about how a vector space can be written as a direct sum of subspaces, where each subspace corresponds to a primary factor of the minimal polynomial of a linear operator. Is this linear algebra version somehow a specific case of the general one, or is it called so just because it depends on a primary decomposition?

Answer 1

Let $f$ be a linear endomorphism of the finite dimensional vector space $V$ defined over the field $k$, $k[X]$ acts on $V$ by $Xu=f(u)$, $V$ is isomorphic to $k[X]/p(X)$ where $p$ is the characteristic polynomial of $f$. Write $p=\Pi_ip_i^{n_i}$, you obtain that $k[X]/p$ is isomorphic to $\bigoplus_ik[X]/(p_i^{n_i})$, remark that $k[X]/(p_i^{n_i})$ is isomorphic to a component of the primary decompostion of $V$ since we have $V=\bigoplus_i\ker(p_i^{n_i}(f))$. The ideal $(p)$ generated by $p$ is the intersection $\bigcap_i(p_i^{n_i})$.