First uniqueness theorem for primary decomposition (${\mathrm{Ass}}_R(R/I)=\{\sqrt{Q_i}\}$)

The distinct radicals in a minimal primary decomposition are exactly the associated primes of a module $R/I$, and hence are invariant. The injection into the sum of primary quotients proves one inclusion; an element supported on one component, supplied by irredundancy, proves the other. This does not imply uniqueness of embedded primary components.