On finite generatedness of projective modules over rings with polynomial identities (Q1901964)

Let \(R\) be an associative ring with identity and \(M_n (R)\) the ring of \(n \times n\) matrices over \(R\). Main results: (i) Let \(I\) be a right ideal of \(R\) and \(P\) a projective \(R\)-module such that \(P/PI\) is finitely generated. If \(I\) is a PI-ring and \(I \subseteq J(R)\) then \(P\) is a finitely generated \(R\)-module. (ii) Let \(R\) be a PI-ring and for any sequence of right ideals \(A_m M_n (R)\) where \(A_m=A_{m+1} A_m\), \(m =1,2,\dots\), there exists a right ideal \(A_p M_n (R)\) containing a just nonidempotent element. Then the following conditions are equivalent: (1) Let \(J_k=\sum^n_{i=1} u_{ik}R\), where \((u_{1m}, \dots, u_{nm})=(u_{1m+1}, \dots, u_{nm+1}) A_m\), and the matrices \(A_m\), \(m=1,2,\dots\), satisfy the condition \(A_m=A_{m+1} A_m\). Then the ascending chain \(J_1 \subseteq J_2 \subseteq \cdots\) is stationary. (2) Let \(P\) be a submodule of a free module \(F=\sum^n_{i=1} x_i R\) and \(P\) be generated by the elements \(\{u_{ij}\}\), where \((u_{1m}, \dots, u_{nm})=(x_1, \dots, x_n) A_m\), \(m =1,2, \dots\), and the set of matrices \(\{A_1, A_2, \dots\}\) satisfies the condition \(A_m=A_{m+1} A_m\). Then \(P\) is finitely generated.