Necklace rings and their radicals (Q2770660)

The notion of necklace ring \(N_k(A)\), \(k=1,2,\dots,\omega\), is extended to an arbitrary ring \(A\), and as a prerequisite the relationship between the ideals of \(A\) and \(N_k(A)\) is discussed. For a radical \(\alpha\) for which the ring of integers is semisimple and for every ring \(A\), \(\alpha(N_k(A))=\alpha(N_k(A^*))\) where \(A^*\) is the Dorroh extension of \(A\). If \(\alpha\) has the matrix extension property then \(\alpha(N_k(M_u(A)))\simeq M_u(\alpha(N_k(A)))\) where \(M_u(X)\) denotes the \(u\times u\) matrix ring over \(X\). It is shown that there is no canonical way to describe the radical of the necklace ring in terms of the radical of the base ring; the relationship depends on the type of the radical (hypernilpotent or hypoidempotent), on the type of ring and on \(k\). Let \(I\triangleleft A\) and \(D_k(I)=\{(a_1,a_2,\dots)\in N_k(A):na_n\in I\text{ for all suitable }n\}\). The equality \(\alpha(N_k(A))=D_k(\alpha(A))\) does not hold in general for supernilpotent radicals, but \({\mathfrak J}(N_k(A))=D_k({\mathfrak J}(A))\) for the Jacobson radical \(\mathfrak J\). The description of hypoidempotent radicals \(\alpha\) of \(N_k(A)\) is less orderly.