Let S(RG) be a normed Sylow p-subgroup in a group ring RG of an abelian group G with p-component Gp and a p-basic subgroup B over a commutative unitary ring R with prime characteristic p. The first central result is that 1 + I(RG; Bp) + I(R(p i )G; G) is basic in S(RG) and B[1 + I(RG; Bp) + I(R(p i )G; G)] is p-basic in V (RG), and [1 + I(RG; Bp) + I(R(p i )G; G)]Gp/Gp is basic in S(RG)/Gp and [1 + I(RG; Bp) + I(R(p i )G; G)]G/G is p-basic in V (RG)/G, provided in both cases G/Gp is p-divisible and R is such that its maximal perfect subring R p i has no nilpotents whenever i is natural. The second major result is that B(1 + I(RG; Bp)) is p-basic in V (RG) and (1 + I(RG; Bp))G/G is p-basic in V (RG)/G, provided G/Gp is p-divisible and R is perfect. In particular, under these circumstances, S(RG) and S(RG)/Gp are both starred or algebraically compact groups. The last results offer a new perspective on the long-standing classical conjecture which says that S(RG)/Gp is totally projective. The present facts improve the results concerning this topic due to Nachev (Houston J. Math., 1996) and others obtained by us in (C. R. Acad. Bulg. Sci., 1995) and (Czechoslovak Math. J., 2002).
An explicit representation for ideal CR submanifolds of a complex hyperbolic space has been derived in T. Sasahara (2002). We simplify and reformulate the representation in terms of certain Kähler submanifolds. In addition, we investigate the almost contact metric structure of ideal CR submanifolds in a complex hyperbolic space. Moreover, we obtain a codimension reduction theorem for ideal CR submanifolds in a complex projective space.