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A ring is (weakly) nil clean provided that every element in is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let be abelian, and let . We prove that is nil clean if and only if is Boolean and is nil. Furthermore, we prove that is weakly nil clean if and only if is periodic; is , or where is a Boolean ring, and that is weakly nil clean if and only if is nil clean for all .
Let be a finite group with a normal subgroup such that . It is shown that under some conditions, Coleman automorphisms of are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.
A new class of abelian -groups with all high subgroups isomorphic is defined. Commutative modular and semisimple group algebras over such groups are examined. The results obtained continue our recent statements published in Comment. Math. Univ. Carolinae (2002).
Let be a -mixed abelian group and is a commutative perfect integral domain of . Then, the first main result is that the group of all normalized invertible elements is a -group if and only if is a -group. In particular, the second central result is that if is a -group, the -algebras isomorphism between the group algebras and for an arbitrary but fixed group implies is a -mixed abelian -group and even more that the high subgroups of and are isomorphic, namely, . Besides,...
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