327 research outputs found
Isomorphism of Commutative Group Algebras over all Fields
It is argued that the commutative group algebra over
each field determines up to an isomorphism its group basis for
any of the following group classes:
⢠Direct sums of cocyclic groups
⢠Splitting countable modulo torsion groups whose torsion parts
are direct sums of cyclics;
⢠Splitting groups whose torsion parts are separable countable
⢠Groups whose torsion parts are algebraically compact
⢠Algebraically compact groups
These give a partial positive answer to the R.Brauerâs classical
problem
Rings with Jacobson units
We introduce and study the notion of JU rings, that are, rings having only Jacobson units. In parallel to the so-called UU rings, these rings also form a large class and have many interesting properties established in the present paper. For instance, it is proved that any exchange JU ring is semi-boolean, and vice versa. This somewhat extends a result due to Lee-Zhou (Glasg. Math. J., 2008) and Danchev-Lam (Publ. Math. Debrecen, 2016)
On the trivial units in finite commutative group rings
Let G be a finite abelian group and F a finite field. A criterion is found for all units in the group ringFG to be trivial. This attainment is also extended to the general case for arbitrary abelian groups and fields
On exchange Ď-UU unital rings
We prove that a ring R is exchange 2-UU if, and only if, J(R) is nil and R/J(R)â
BĂC, where B is a Boolean ring and C is a ring with C â Î Îź â¤â for some ordinal Îź. We thus somewhat improve on a result due to Abdolyousefi-Chen (J. Algebra Appl., 2018) by showing that it is a simple consequence of already well-known results of Danchev-Lam (Publ. Math. Debrecen, 2016) and Danchev (Commun. Korean Math. Soc., 2017)
Basic subgroups in abelian group rings
summary: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. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring is perfect and is -primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup
Warfield Invariants of
Let be a commutativeunitary ring of prime characteristic and let be an Abelian group. We calculateonly in terms of and (and their sections)Warfield -invariants of thequotient group , that is, the group of all normalized units in the groupring modulo . This supplies recent results of ours in (Extr. Math., 2005),(Collect. Math., 2008) and (J. Algebra Appl., 2008)
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