×

\(\mathcal{C}\)-resolutions and \(\mathcal{C}\)-dimensions. (Chinese. English summary) Zbl 1474.16029

Summary: Let \(R\) be an associative ring with identity, and \(\mathcal{C}\) be a class of left \(R\)-modules which contains all injective left \(R\)-modules. A left \(R\)-module \(M\) is \(\mathcal{C}\)-injective (reduced \(\mathcal{C}\)-injective) if and only if \(M\) is a kernel of \(E \to L\) which is a \(\mathcal{C}\)-precover (\(\mathcal{C}\)-cover) of a left \(R\)-module \(L\) with \(E\) injective. If the ring \(R\) is about kernels of \(\mathcal{C}\)-precovers factorable through injective modules, the relations of left \(\mathcal{C}\)-resolution, left \(\mathcal{C}\)-dimension and derived functors are characterized, and some applications are given.

MSC:

16E05 Syzygies, resolutions, complexes in associative algebras
16E10 Homological dimension in associative algebras
PDFBibTeX XMLCite