Decomposition of a compact ring into the sum of the radical and an inertial subring
Let A be a compact alternative or Jordan ring with 1. Denote \(K=K(A)\) the closed subring generated by 1. Suppose that A/J(A) is a simple and finite ring of characteristic \(p>0\). Here J(A) is the quasiregular radical of A. The author proves that K is a Hensel ring. There exists a closed subring \(A_ 1\) containing 1 such that \(A=J(A)+A_ 1\), \(J(A_ 1)=p\cdot A\) and \(K(A_ 1)=K\). Moreover, \(A_ 1\) is an algebra belonging to the following types: (1) the matrix ring \(L_ n\), where L is the center of \(A_ 1\); (2) the split Cayley-Dickson algebra over the center; (3) the Jordan algebra of a symmetric bilinear form; (4) the algebra \(H(M_ n,J_ a)\), \(n\geq 3\), and the author gives a complete description of all possible types of the alternative ring M with involution j. This result is analogous to results obtained for associative algebras by \textit{G. Azumaya} [Nagoya Math. J. 2, 119-150 (1951; Zbl 0045.011)], for associative compact rings by \textit{Z. Lipkina} [Sib. Mat. Zh. 14, 1346- 1348 (1973; Zbl 0287.16024)] and for alternative and Jordan algebras by \textit{V. N. Zhelyabin} [Tr. Inst. Mat. 4, 5-28 (1984; Zbl 0559.17011)]. The last one was essentially used here.
scientific article; zbMATH DE number 4095720
scientific article