The equivariant real, complex and quaternionic vector fields on spheres problem is reduced to a question about the equivariant J -groups of the projective spaces. As an application of this reduction, we give a generalization of the results of Namboodiri [U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. 278 (2) (1983) 431-460], on equivariant real vector fields, and Ön-der [T. Önder, Equivariant cross sections of complex Stiefel manifolds, Topology Appl. 109 (2001) 107-125], on equivariant complex vector fields, which avoids the restriction that the representation containing the sphere has enough orbit types.

Let KO(CP m ) be the KO-ring of the complex projective space CP m . By means of methods of rational D-series [4], a formula for the J-orders of elements of KO(CP m ) is given. Explicit formulas are given for computing the J-orders of the canonical generators of KO(CP m ) and the J-order of any complex line bundle over CP m .

A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard (4n − 1)-sphere is shown to be twice the number of linearly independent quaternionic vector fields plus d, where d = 1 or 3.

A formula for the number of gradations, up to equivalence, of cyclic rings by cancellative monoids is given. As an application, the nil and Jacobson radicals of cyclic rings are shown to be homogeneous.

Let JO(X) = KO(X)/T O(X) be the J-group of a connected finite CW complex X. Using Atiyah-Tall [5], we obtain two computable formulae of T O(X) (p) , the localization of T O(X) at a prime p. Then we show how to use those two formulae of T O(X) (p) to find the J-orders of elements of KO(X), at least the 2 and 3 primary factors of the canonical generators of JO(CP m ). Here CP m is the complex projective space. 1991 Mathematics Subject Classification: Primary 55Q50, 55R50.