Definition 7

Let A be an algebra over K. A primitive ideal of A is defined to be the kernel of a nontrivial irreducible representation of A.

If A is commutative, the primitive ideals of A are the regular maximal ideals of A. In fact, the nontrivial irreducible representations of A are, up to equivalence, the representations $\pi_R$ defined by the A-pseudomodules $A/R$, where R is a regular maximal ideal of A. The kernel of $\pi_R$ contains R. It is even equal to it since, by A, VIII, p. 426, Prop. 2, the commutativity of A implies that $A/R$ is a field. Hence Ker($\pi_R$) is regular maximal.