On the unit groups of rings with involution

Subnormal subgroups of almost locally simple artinian algebras with involutions

“…The main result of this section is Theorem 4.9 in which we give some equivalent conditions on the sets of traces T G and norms N G , where G is a non-central subnormal subgroup of GL n (D), and these conditions, in some sense, are very strong implying a special construction for the whole M n (D). It can be seen that Theorem 4.9 broadly extends [5,Theorem 6.1]. In Section 5, we try to extend the results obtained in Section 4 for the class of almost locally simple artinian algebras.…”

“…This paper is a sequel of [5], where the structures of subgroups of certain algebras with involutions were investigated. Let G be a subgroup of the multiplicative subgroup A × of a ring A with involution ⋆.…”

“…In [5], instead of considering the two sets T A and N A , the authors have considered much smaller subsets T G and N G , where G is a non-central normal subgroup of A × . The main results of [5] demonstrates that the sets T G and N G are too big in the whole A; that is, if we impose suitable conditions on T G ∪ N G , then the algebraic structure of the whole ring A is effected.…”

“…In [5], instead of considering the two sets T A and N A , the authors have considered much smaller subsets T G and N G , where G is a non-central normal subgroup of A × . The main results of [5] demonstrates that the sets T G and N G are too big in the whole A; that is, if we impose suitable conditions on T G ∪ N G , then the algebraic structure of the whole ring A is effected. More precisely, it was shown that if A is a division ring and T G ∪ N G is commutative, then A is a quaternion division algebra over Z(A) and ⋆ is of the symplectic type (see [5,Theorem 5.5]).…”

“…The main results of [5] demonstrates that the sets T G and N G are too big in the whole A; that is, if we impose suitable conditions on T G ∪ N G , then the algebraic structure of the whole ring A is effected. More precisely, it was shown that if A is a division ring and T G ∪ N G is commutative, then A is a quaternion division algebra over Z(A) and ⋆ is of the symplectic type (see [5,Theorem 5.5]). Moreover, if A is a simple artinian ring and T G ∪N G is contained in Z(A), then either A is a quaternion division algebra over Z(A) or A is isomorphic to the matrix ring M 2 (Z(A)) and ⋆ is of the symplectic type (see [5,Theorem 6.1]).…”