The basic theme of this paper, suggested by Ottmar Loos, is to show that certain setsSofelements in a Jordan systemJform an ideal by showing thatS=J∩R([Jtilde]) is the Amitsur shrinkage of some well-knownl radicalRon some extension[Jtilde]ofJ. The Jacobson radical Rad(J) and the degenerate radical Deg(J) have elemental characterizations as (respectively) the properly quasi-invertible elements and the m-finite elements. Frojn these two characterizations we show that: (1) the strictly properly nilpotent elements coincide with the strictly properly quasi-invertible elements and form the idealJ∩Rad(J[T]) (2) the strictly m-finite elements coincide with the m-finite elements and form the idealDeg:(J) (3) the m-bounded elements form an idealJ∩Deg(Seq(J)) (Seq the algebra of sequences); and (4) the strictly m-bounded elements coincide with the strictly properly nilpotence-bounded elements and form the idealJ∩Deg(Seq(J[t])). We show that all these constructions are stable under structural pairs, a useful generalization of the concept of structural transformation. The question of whether the properly nilpotent elements form an ideal, and if so whether this is the nil radical, is an open question intimately related to the Kothe Conjecture for associative algebras.