Almost hereditary rings
If \(R\) is a right artinian ring, it is well known that \(R\) is right hereditary if and only if the Jacobson radical of \(R\) is projective as a right \(R\)-module. In view of this, the author calls a (basic) right artinian ring \(R\) (right) almost hereditary if the Jacobson radical of \(R\) is almost projective as a right \(R\)-module. If \(R\) is a hereditary ring, it follows that every submodule of a finitely generated projective right \(R\)-module is projective. However, this is not the case for almost hereditary rings. That is, when \(R\) is almost hereditary, it does not follow that submodules of finitely generated projective right \(R\)-modules are almost projective. The author presents two main theorems in the paper. He shows that: 1) \(R\) is almost hereditary if and only if \(R\) is a direct sum of i) hereditary rings, ii) serial rings (= two sided Nakayama rings) and iii) special triangular matrix rings over hereditary rings and serial rings of the first category. A ring is in the first category if it is a factor ring of a triangular matrix ring of a certain type whose entries are taken from a division ring; and 2) \(R\) is a two sided almost hereditary ring if and only if \(R\) is a direct sum of hereditary rings and serial rings. The author also investigates and obtains interesting results on the structure of the following rings: 1) Basic artinian rings for which every submodule of each finitely generated projective module is almost projective; 2) Basic artinian rings such that the Jacobson radical of every finitely generated almost projective module is again almost projective; and 3) Basic artinian rings for which every submodule of each almost projective module is almost projective.
finitely generated projective right \(R\)-module
scientific article; zbMATH DE number 91624