On direct decompositions of \(p\)-adic groups (Q2761488)

A ring \(R\) is semilocal if \(R/J\) is Artinian where \(J\) is the Jacobson radical. The author proves the existence of a \(p\)-local torsion free Abelian group of finite rank whose endomorphism ring is a special semilocal ring \(R\): the direct decomposition of modules appearing as a factor of finitely generated projectives by the Jacobson radical is prescribed. Moreover, the orders of the matrix algebras in the decomposition of \(R/J\) have prescribed size. All the prescriptions are arbitrary subject to some obviously necessary restrictions. This is an extension of the main result of \textit{A. Facchini} and \textit{D. Herbera} [J. Algebra 225, No. 1, 47-69 (2000; Zbl 0955.13006)], which states the existence of semilocal rings with prescribed direct decompositions.NEWLINENEWLINENEWLINEThe proof relies on some unusual additive categories of Abelian groups whose theory was developed by the author [in Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 175, 135-153 (1989; Zbl 0745.20050)].