LOCAL UNITS AND CIRCULAR INDEX IN ABELIAN P-GROUP RINGS

Let A be a finite abelian group of exponent p(m) > 1, an odd prime power, and consider the Z(p)-module DELTA(p)+ (A) generated by the set {(z - 1) + (z-1 - 1)\z is-an-element-of A} in the group ring Z(p)A. We show that the multiplicative group S(p)(A) = 1 + DELTA(p)+ (A) has a Z(p)-basis of the form {w(p)(z)\1 not-equal z is-an-element-of A}, where w(p)(x) is a fixed Z(p)-polynomial. The integral unit group S(A) = (ZA)x and S(p)(A) is known to have a canonical subgroup LAMBDA(A) of a similar structure; it lies in the group OMEGA(A) of all units appearing circular under every character of A. Comparing the two situations, we conclude that the index [OMEGA(A) : LAMBDA(A)] equals the order of a certain group of ideal classes of ZA.