The next meeting of the Algebra and Logic Seminar will be held
DECOMPOSING NORMALIZED UNITS IN COMMUTATIVE MODULAR GROUP RINGS
Abstract
Introduction: Suppose G is an abelian p-group and F is a perfect field of characteristic p > 0. As usual, FG denotes the group algebra of G over F with group of normalized units V(FG) and augmentation ideal I(FG;G). For any ideal I of FG, we set I^n as the product of I written n times, where n is an arbitrary fixed natural number. The numerous decompositions of the normed group V(FG), in which the former group G can be viewed as a decomposing component, are currently of some interest and significance in the theory of commutative modular group algebras.
Actuality: This subject is closely related to the so-called Direct Factor Problem (DFP) and the statements obtained below shed some more light in that way. Specifically, the DFP asked of whether or not the group G is always a direct factor of V(FG) with totally projective complement.
Results: We prove the following two, somewhat curious theorems:
T1. Let F be a perfect field of characteristic p and let G be an abelian p-group. Then the equality V(FG) = G(1+I^2(FG;G)) holds if, and only if, G is divisible, that is, G = G^p, or G is not divisible and F is the p-element field Z_p.
In particular, the equality V(Z_pG) = G(1+I^2(Z_pG;G)) is always fulfilled, provided G is an abelian p-group.
T2. Let F be a perfect field of characteristic p > 2 and let G be an abelian p-group. Then the equality V(FG) = G(1+I^p(FG;G)) holds if, and only if, G is divisible, i.e., G = G^p.
Methods: We use technique which exploits certain decompositions of units in modular group rings as well as some special homomorphisms between linear spaces.
An open problem: Find a criterion only in terms of F and G for the validity of the equality V(FG) = G(1+I^n(FG;G)), where n is an arbitrary but fixed natural.
Publishing status: These results are an important part of a subsequent own publication of Peter V. Danchev under the same title as above in the journal/periodical of the Romanian Academy of Sciences “Rev. Rouman. Math. Pures & Appl.”, vol. 64, No 1, 2019.