Nó được xây dựng một vòng đặc biệt R giao hoán đơn nhất của đặc trưng 2, mà không nhất thiết yếu hoàn hảo (vì thế không hoàn hảo) hoặc đếm được, và nó được chọn một abelian nhân giống 2-nhóm G là một tổng trực tiếp của. | Vietnam Journal of Mathematics 34 3 2006 265-273 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 Simply Presented Inseparable V RG Without R Being Weakly Perfect or Countable Peter Danchev 13 General Kutuzov Str. block 7 floor 2 flat 4 4003 Plovdiv Bulgaria Received July 06 2004 Revised June 15 2006 Abstract. It is constructed a special commutative unitary ring R of characteristic 2 which is not necessarily weakly perfect hence not perfect or countable and it is selected a multiplicative abelian 2-group G that is a direct sum of countable groups such that V RG the group of all normed 2-units in the group ring RG is a direct sum of countable groups. So this is the first result of the present type which prompts that the conditions for perfection or countability on R can be probably removed in general. 2000 Mathematics Subject Classification 16U60 16S34 20K10. Keywords Unit groups direct sums of countable groups heightly-additive rings weakly perfect rings. Let RG be a group ring where G is a p-primary abelian multiplicative group and R is a commutative ring with identity of prime characteristic p. Let V RG denote the normalized p-torsion component of the group of all units in RG. For a subgroup D of G we shall designate by I RG D the relative augmentation ideal of RG with respect to D that is the ideal of RG generated by elements 1-d whenever d G D. Warren May first proved in 11 that V RG is a direct sum of countable groups if and only if G is provided R is perfect and G is of countable length. More precisely he has argued that if G is an arbitrary direct sum of countable groups and R is perfect V RG G and V RG are both direct sums of countable groups for their generalizations see 12 and 2 8 as well . At this stage even if the group G is reduced there is no results of this 266 Peter Danchev kind which are established without additional restrictions on the ring R. These restrictions are perfection R Rp weakly perfection Rp Rp for some i G N and countability R Ko . .
