By Donald S. Passman

Passman D.S. endless crossed items (AP, 1989)(ISBN 0125463901)

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**Example text**

Show that a is a central element of K[G]satifying a(a - IN/) = 0. Conclude that K[G]is not prime and that K[G] is not semiprime if (NI = 0 in K . 5. 4. Note that A/A+ is a torsion-free abelian group and hence an ordered group. 5. Suppose R is a simple ring, u is an automorphism of R and 0 # u E R. If ur = r‘u for all T E R, show that u is a unit of R and 0 is the inner automorphism of R induced by the unit u-’. 6. Let R*G be given with R a simple ring and assume that for all 1 # z E G the automorphism ” is outer on R.

0 and therefore that T A ( A ). T A ( B )= 0. This reduces the various questions to K[A] where they are easily solved. 3. If char K = 0, then K[G] is semiprime. 4. [155] If charK and only if Ap(G) = (1). =p > 0, then K[G] is semiprime if Here AP(G) is the subgroup of A+(G) generated by all elements of order a power of p . 5. [44] K[G] is prime if and only if A+(G) = (1). Now let us move on to crossed products R*G. As will be apparent, when G acts nontrivially on the ring R, a new dimension is added to the problem.

Thus we must take a different approach. Suppose first that L n ziA = 0 for all i = 1 , 2 , . . Then KhK = 0 for all h with h E ( H \ S ) u (H\ L)= H \ (Ln S ) and by the above assumption, L n S = LnT is a finite union of cosets of the groups B = L n A I A E d'}. Since 1231 5 ld'l < Id1 and { 52 2. Delta Methods and Semiprime Rings since B is clearly closed under intersections, induction applies here. Thus, as before, there exists I = K U 1 K U* 2- KuB# 0 with some uj = 1 such that IhI = 0 for all h E H \ L where IL : r l BI < 00 for some B E B.