Semihereditary semichain rings (Q1094516)

The authors prove that any semichain semihereditary ring A is equivalent to a direct product of cell-triangular matrix rings \[ k= \begin{pmatrix} A_ 1 & M_{n_ 1 \times n_ 2}(D) & ... & M_{n_ 1 \times n_ t}(D) \\ 0 & A_ 2 & ... & M_{n_ 2 \times n_ t}(D) \\ 0 & 0 & ... & ... \\ 0 & 0 & 0 & A_ t \end{pmatrix} \] in the sense of Morita contexts. Rings \(A_ 1,...,A_ t\) are prime, semichain and semihereditary. The ring of quotients \(\tilde A_ i\) for \(A_ i\) is isomorphic to some matrix ring \(M_{n_ i}(D)\), \(i=1,...,t\) over a skew field D. Proving this theorem the authors use two helpful results: (i) any semiprime semichain ring is a direct product of prime rings; (ii) any semichain ring has a classical ring of quotients.