\(\lambda\) and \(\mu\)-dimensions of modules. (Q2568710)

If \(\mathcal F\) is a class of \(R\)-modules closed under finite direct sums, then, by using \(\mathcal F\)-(pre)covers, the authors introduce \(\lambda\)- and \(\mu\)-dimensions of an \(R\)-module. A morphism \(\varphi\colon F\to M\) is called a precover of \(M\), if \(F\) is in \(\mathcal F\) and the sequence \(\Hom(F',F)\to\Hom(F',M)\to 0\) is exact for all \(F'\in{\mathcal F}\). If any morphism \(f\colon F\to F\) such that \(\varphi=\varphi\circ f\) is an automorphism of \(F\), then \(\varphi\) is an \(\mathcal F\)-cover of \(M\). Then \(\lambda(M)=-1\), if \(M\) does not have an \(\mathcal F\)-precover; \(\lambda(M)=n\), if there is a partial left resolution of \(M\) of length \(n\) and there are no longer such complexes; \(\lambda(M)=\infty\), if there are \(\mathcal F\)-resolutions of length \(n\) for all natural \(n\). A study of the properties of \(\lambda(M)\) is done. So, \(\lambda(M\oplus F)=\lambda(M)\), for all \(F\) in \(\mathcal F\); if \(\lambda(M)\geq n>k\geq 0\) and \(F_k\to F_{k-1}\to\cdots\to F_0\to M\to 0\) is a partial left \(\mathcal F\)-resolution of \(M\), \(K=\text{Ker}(F_k\to F_{k-1})\) and \(F_{-1}=M\), then \(\lambda(K)\geq n-k-1\); if \(\lambda(M)=n\), then \(\lambda(K)=n-k-1\). If \(M\to N\) is a linear map such that the induced \(\Hom(F,M)\to\Hom(F,N)\) is an isomorphism for all \(F\) in \(\mathcal F\), then \(\lambda(M)=\lambda(N)\). For the other concept of dimension (which inside the paper is denoted by a \(\lambda\) bar), a special \(\mathcal F\)-cover is used, which is nice in the case of the class of Gorenstein projective \(R\)-modules.