Annihilator (ring theory)

The annihilator records which ring elements kill a specified subset of a module.

The right annihilator of a subset $X$ of a ring is the right ideal of elements killed on the left by every member of $X$. Its sidedness matters for a noncommutative ring.

An associated prime of an $R$-module $M$ is a prime ideal equal to the annihilator ${\mathrm{Ann}}_R(m)$ of some nonzero $m\in M$. The set of these primes is denoted ${\mathrm{Ass}}_R(M)$. Over a Noetherian ring, any maximal member among the annihilators of nonzero elements is prime: if $abm=0$ and $bm\ne 0$, maximality forces ${\mathrm{Ann}}_R(bm)={\mathrm{Ann}}_R(m)$, so $am=0$. Hence every nonzero module has an associated prime.

For a commutative Noetherian ring, an annihilator $Q={\mathrm{Ann}}_R(m)$ contained in $P$ localizes to the annihilator of the nonzero $m/1$. Conversely, contract an associated prime of $M_P$ represented by $m/1$ to an ideal $Q$. Choose finitely many generators of $Q$ and one denominator outside $P$ clearing all their annihilation equations. For this denominator $s$, the element $sm$ is nonzero and has annihilator exactly $Q$. This proves the formula; finite generation of the contracted ideal is the key use of Noetherianity.

// Destination: noncommutative-algebra.bigb

An associated prime of a module $R/I$ is embedded when it is not minimal over $I$. For example, in $k[x,y]/(x^2,xy)$ the nonzero class of $x$ has annihilator $(x,y)$, while the unique minimal prime is $(x)$. Embedded primes record annihilators which are invisible if one retains only the reduced irreducible components.

For a Noetherian ring and a proper ideal $I$, every minimal prime $P$ over $I$ is an associated prime of a module $R/I$. The localized quotient has one prime, so its finitely generated maximal ideal is nilpotent and its socle is nonzero. An element with annihilator $P{R}_P$ can be lifted to the quotient. Clearing denominators for a finite generating set of $P$ gives a nonzero element with annihilator exactly $P$.

The annihilator ${\mathrm{Ann}}_R(M)$ is the ideal of scalars that kill every element of $M$.

Over a commutative unital ring, an ideal maximal among annihilators of nonzero elements of a module is a prime ideal. Each such ideal is the annihilator of a module given by the cyclic submodule generated by that element. If $P=\mathrm{Ann}(m)$ and $ab\in P$ with $b\notin P$, then $bm\ne 0$ and $P\subseteq \mathrm{Ann}(bm)$. Maximality makes these annihilators equal, so $a\in P$. A Noetherian ring ensures that a nonzero module has a maximal element annihilator by the ascending chain condition on ideals.