A generalization of heredity ideals (Q2639938)

Let A be a left and right Artinian ring, \(\{e_ 1,...,e_ n\}\) a complete set of orthogonal primitive idempotents in A, \(c_{ij}\) the composition length of \(_{e_ iAe_ i}e_ iAe_ j\) and \(C(A)=(c_{ij})\) the left Cartan matrix of A. The main result is the following theorem which provides a method to reduce the size of the matrix C(A) when the open problem [gl dim A\(<\infty \Rightarrow \det C(A)=1?]\) is discussed. Theorem. Let Q be a torsionless left A-module, I its trace ideal and the following conditions are satisfied: (a) \(D=End_ A(Q)\) is a division ring; (b) the evaluation map \(Q\otimes_ DHom_ A(Q,A)\to A\) is monic; (c) \(Tor^ A_ k(Tr Q,Q)=0\) for \(k\geq 2\) where Tr is the transpose; (d) proj dim\({}_ AQ<\infty.\) Then we have: (1) gl dim A/I\(\leq gl \dim A+proj \dim_ AQ;\) (2) gl dim \(A\leq gl \dim A/I+\max \{2,proj \dim_ AQ+1\};\) (3) det C(A/I)\(=\det C(A).\) (The last condition is a generalization of heredity ideals.) If gl dim A\(<\infty\) then there exists a torsionless left A-module which satisfies the conditions (a) and (b) in the Theorem.