Strongly \(J\)-\(n\)-coherent rings (Q6595655)

"Let \(R\) be a ring. Recall that the Jacobson radical of \(R\), denoted by \(J\), is the intersection of the maximal left (or right) ideals of \(R\). A submodule \(S\) of an \(R\)-module \(M\) is said to be \textit{small} (or \textit{superfluous}) in \(M\) if whenever \(N\) is a submodule of \(M\) such that \(S+N=M\), then \(N= M\). A right ideal of \(R\) is small if it is small when viewed as a submodule of \(R_R\). It is well-known that a right ideal \(A\) is small in \(R\) if and only if \(A \subseteq J\).\N\NIn the paper under review, the author introduces and studies the following concepts: For a fixed positive integer \(n\), a right \(R\)-module \(M\) is called\N\begin{itemize}\N\item \textit{strongly \(J\)-\(n\)-injective} if every \(R\)-homomorphism from an \(n\)-generated\Nsmall submodule of a free right \(R\)-module \(F\) to \(M\) extends to a homomorphism\Nof \(F\) to \(M\);\N\N\item \textit{strongly \(J\)-\(n\)-flat}, if for every \(n\) generated small submodule \(T\) of a free left \(R\)-module \(F\), the canonical map \(M\otimes T \to M \otimes F\) is a monomorphism.\N\end{itemize}\NFor a fixed positive integer \(n\), a ring R is called\N\begin{itemize}\N\item \textit{left strongly \(J\)-\(n\)-coherent} if every \(n\)-generated small submodule of a free left \(R\)-module is finitely presented; \N\N\item \textit{left \(J\)-\(n\)-semihereditary} if every \(n\)-generated small left ideal of \(R\) is projective.\N\end{itemize}\NThen the author proves the following results (only a few of them are stated below):\N\begin{itemize}\N\item[1.] \(M_R\) is strongly \(J\)-\(n\)-injective if and only if \(Ext^1(F/T,M) = 0\) for every finitely generated free right \(R\)-module \(F\) and every \(n\)-generated small submodule \(T\) of \(F\) (Theorem 2.4).\N\N\item[2.] \(M_R\) is strongly \(J\)-\(n\)-flat if and only if \(Tor_1(M,F/T) = 0\) for every finitely generated free left \(R\)-module \(F\) and any \(n\)-generated small submodule \(T\) of \(F\) (Theorem 3.2).\\\N\N\item[3.] \(R\) is left strongly \(J\)-\(n\)-coherent if and only if given an exact sequence of left \(R\)-modules\N\[\N\begin{tikzcd}\N0 \arrow[r] & K \arrow[r, ""f""] & R^n \arrow[r, ""g""] & T\N\end{tikzcd}\N\]\Nwith \(T\) is a finitely generated small submodule of a free left \(R\)-module, the module \(K\) is finitely generated.\N\NEquivalently, \(\varinjlim Ext^1(F/T,M_{\alpha})\cong Ext^1(F/T,\varinjlim M_{\alpha})\) for every \(n\)-generated small submodule \(T\) of a finitely generated free left \(R\)-module \(F\) and direct system\N\((M_{\alpha})_{{\alpha}\in A}\) of left \(R\)-modules (Theorem 4.4 and Theorem 4.5).\N\N\item[4.] \(R\) is a left \(J\)-\(n\)-semihereditary ring if and only if \(R\) is left strongly \(J\)-\(n\)-coherent and every submodule of a strongly \(J\)-\(n\)-flat right \(R\)-module is strongly \(J\)-\(n\)-flat.\\\indent Equivalently, every quotient module of a strongly \(J\)-\(n\)-injective left \(R\)-module is strongly\N\(J\)-\(n\)-injective (Theorem 5.4).\N\end{itemize}"