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Let be the set of nonnegative integers and the ring of integers. Let be the ring of matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of...
Suppose is a field of characteristic and is a -primary abelian -group. It is shown that is a direct factor of the group of units of the group algebra .
This article presents a brief survey of the work done on rings generated by their units.
We give a comment to Theorem 1.1 published in our paper “Ring elements as sums of units” [Cent. Eur. J. Math., 2009, 7(3), 395–399].
In 1950 N. Jacobson proved that if u is an element of a ring with unit such that u has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky-Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky-Jacobson...
Suppose is a commutative ring with identity of prime characteristic and is an arbitrary abelian -group. In the present paper, a basic subgroup and a lower basic subgroup of the -component and of the factor-group of the unit group in the modular group algebra are established, in the case when is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed -component and of the quotient group are given when is perfect and is arbitrary whose is -divisible....
Let be a normed Sylow -subgroup in a group ring of an abelian group with -component and a -basic subgroup over a commutative unitary ring with prime characteristic . The first central result is that is basic in and is -basic in , and is basic in and is -basic in , provided in both cases is -divisible and is such that its maximal perfect subring has no nilpotents whenever is natural. The second major result is that is -basic in and is -basic in ,...
Suppose is a perfect field of and is an arbitrary abelian multiplicative group with a -basic subgroup and -component . Let be the group algebra with normed group of all units and its Sylow -subgroup , and let be the nilradical of the relative augmentation ideal of with respect to . The main results that motivate this article are that is basic in , and is -basic in provided is -mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston...
We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a matrix over a projective-free ring is strongly -clean if and only if , or , or is similar to , where , , and the equation has a root in and a root in . We further prove that is strongly -clean if be optimally -clean.
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