= Primitive ideal
{title2=$P=\operatorname{Ann}_A(E),\quad E\text{ simple}$}
A primitive ideal of a unital <associative algebra> is the <annihilator of a module> for a nonzero <simple module>. Every maximal proper two-sided ideal is primitive: its quotient is a simple ring, and any simple module over that quotient has zero annihilator there. In a unital <Banach algebra>, a simple module can be realized as $A/L$ for a <maximal left ideal>; this ideal is closed, making the representation a bounded algebraically irreducible action on a <Banach space>.
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