Full Ideals and Ring Groups in Zn[x]

If we add the operation of composition to the polynomial ring R[X], where R is a commutative ring with identity, we get a tri-operational algebra . A full ideal or tri-operational ideal of is the kernel of a tri-operational homomorphism on . This is equivalent [4, pp. 73–74] to the following: A full ideal of is a ring ideal A of R[x] such that f°g ∈ A for every f ∈ A and g ∈ R[x].

XVII.—Quaternion Investigation of the Commutative Law for Homogeneous Strains

1. If a plastic body be subjected to a uniform change of shape, any two equal, similarly placed cubes become equal, similarly placed parallelepipeds. It is well known that the character of the deformation may be determined by a set of nine constants; but the physicist naturally prefers to regard the strain as an operator, and to represent it by a single symbol. The resulting operational algebra may even react favourably on the mathematical aspect of the question, and give us new methods of attack for old problems.