On almost precovers and almost preenvelopes. (Q2507621)

Let \(\mathcal C\) be a class of modules closed under isomorphisms and finite direct sums. Recall that a homomorphism \(\varphi\colon X\to M\) with \(X\in\mathcal C\) is called an almost \(\mathcal C\)-precover of \(M\), if for each \(F\in\mathcal C\) and each \(f\colon F\to M\) there is an essential submodule \(F'\leq F\) with \(F'\in\mathcal C\) and a homomorphism \(g\colon F'\to X\) such that \(\varphi g=f\iota\), \(\iota\colon F'\to F\) being the inclusion map. A class \(\mathcal C\) is called weakly hereditary if each non-zero submodule of \(M\in\mathcal C\) contains an essential submodule from the class \(\mathcal C\). If \(\mathcal C\) is weakly hereditary and if \(\varphi_i\colon X_i\to M_i\), \(i=1,2,\dots,n\), are almost \(\mathcal C\)-precovers, then \(\bigoplus_{i=1}^n\varphi_i\) is an almost \(\mathcal C\)-precover of \(\bigoplus_{i=1}^nM_i\) (Theorem 2.2). An almost \(\mathcal C\)-precover \(\varphi\colon G\to M\) is called a weak \(\mathcal C\)-cover if each endomorphism \(f\) of \(G\) with \(\varphi f=\varphi\) is an essential monomorphism (\(\text{Im\,}f\) is essential in \(G\)) and \(\varphi\) is called an almost \(\mathcal C\)-cover if each endomorphism \(f\) of \(G\) with \(\varphi f=\varphi\) is an automorphism of \(G\). If \(\mathcal C\) is weakly hereditary, \(\varphi_1\colon X_1\to M_1\) is an almost \(\mathcal C\)-cover and \(\varphi_2\colon X_2\to M_2\) is a weak \(\mathcal C\)-cover, then \(\varphi_1\oplus\varphi_2\) is a weak \(\mathcal C\)-cover of \(M_1\oplus M_2\). The last paragraph is devoted to the study of almost preenvelopes which are defined dually to the almost precovers.