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Now let Y be the R-module with basis if and only if q~??? and q~??? are if A4 N N as A-modules then Ur We shall now show that &7 is not V as A-- v1 =17x, v2 = 17x2,..., vpu2 = p-2 9 V pp1 =A + q,lIa =Xp-I. Y is R-pure and R is a principal summand of N. K OR V is the augmentation ideal of KG, so K OR VZ K@, U and N/V-R since K@N/VZKGJN/K@ e see that ideal domain, hence V is an R-direct V-KG/K@ U-K.
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