Let A be a nondegenerate noncommutative Jordan algebra over a field of characteristic not two. The socle of A is the sum of the minimal inner ideals of A. The authors prove that the socle contains all elements in A of finite rank, that is, all \(b\in A\) such that \(U_ bA\) is finite- dimensional. Every element of the socle is shown to be von Neumann regular and to have finite spectrum. The annihilator of the socle can be regarded as a radical containing the Jacobson radical. The annihilator of the socle is zero if and only if A is a subdirect sum of primitive nondegenerate noncommutative Jordan algebras having nonzero socles contained in A. If A is Banach, its socle is the maximal regular ideal of A. The preceding results are specialized to alternative algebras.