annihilator

The annihilator of a module records exactly which ring elements act as zero on it — the scalars that flatten the whole module to nothing. It is a precise way to ask 'how much of the ring actually does anything here?'. The bigger the annihilator, the more of the ring's action is wasted, and the more the module really lives over a smaller quotient ring.

For an R-module M, the annihilator is Ann_R(M) = { r in R : rm = 0 for all m in M }; it is a two-sided ideal of R when M is a bimodule, and a left/right ideal in the one-sided case. For a single element m, the annihilator Ann_R(m) = { r : rm = 0 } is a left ideal, and the cyclic submodule Rm is isomorphic to R/Ann_R(m). The annihilator of the whole module is the intersection of the annihilators of its elements.

A module M is naturally a faithful module over the quotient ring R/Ann_R(M), so the annihilator measures the gap between the abstract ring and the part of it that genuinely acts. For finitely generated modules over a commutative Noetherian ring, the radical of the annihilator equals the intersection of the associated primes, tying the annihilator directly to the support and primary decomposition of M. The annihilator of a torsion element generates that element's 'order' over a PID.