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I have extensively studied group rings and structure of their units and also zero divisors and normalizer problem in integral group rings.. I was pondering upon questions like what if we use topological rings under various topologies where either group is group or it can also be topological group. What happens to augmentation ideals and their group of units. Group rings can also be identified as R-modules and Arnautov has studied topological modules in his book. I want to use various kind of compactness or spaces to prepare a research paper on such problems.

If anyone has any link to such papers if they are in literature please post them here. I could only find "On group rings of topological groups" by Iwasawa, Kenkichi which is from 1944 and "Topological Representations of Abelian Group Rings" by Chatzidakis, Z.; Pappas, P.from 1991.

They do not deal with topological rings as my main priority is to use topologies on ring R as topological rings instead of topological groups.

Edit- The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\Bbb{Z}G)$.

I found that this has been proved for finite nilpotent groups , groups of odd order and for groups having a normal sylow 2-subgroup i.e. $2$-closed groups

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  • $\begingroup$ I think, you should explain the meaning of the terms, because this seems to be addressed to specialists in functional analysis, and at least some of them (including me) do not know what is meant by "units", "normalizer problem", etc. And the question itself should be clarifierd, in my opinion. $\endgroup$
    – Sergei Akbarov
    Commented Jul 8, 2022 at 10:25
  • $\begingroup$ In this context, I assume "units" refer to multiplicatively invertible ring elements. Also, give a subgroup $H$ of $G$, one can ask which units $g$ of group ring $RG$ normalize $H$; that is, satisfy $gHg^{-1}$. $\endgroup$
    – Mithrandir
    Commented Jul 8, 2022 at 11:19
  • $\begingroup$ @Mithrandir do you mean $gHg^{-1}=H$? $\endgroup$
    – Sergei Akbarov
    Commented Jul 8, 2022 at 11:24
  • $\begingroup$ @SergeiAkbarov yes, apologies $\endgroup$
    – Mithrandir
    Commented Jul 8, 2022 at 11:36
  • $\begingroup$ And what is meant by "topological ring" of the group? $\endgroup$
    – Sergei Akbarov
    Commented Jul 8, 2022 at 13:33

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