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Some results on \((n,d)\)-injective modules, \((n,d)\)-flat modules and \(n\)-coherent rings. (English) Zbl 1363.16013

Summary: Let \(n,d\) be two non-negative integers. A left \(R\)-module \(M\) is called \((n,d)\)-injective, if \(\text{Ext}^{d+1}(N,M)=0\) for every \(n\)-presented left \(R\)-module \(N\). A right \(R\)-module \(V\) is called \((n,d)\)-flat, if \(\text{Tor}_{d+1}(V,N)=0\) for every \(n\)-presented left \(R\)-module \(N\). A left \(R\)-module \(M\) is called weakly \(n\)-\(FP\)-injective, if \(\text{Ext}^n(N,M)=0\) for every \((n+1)\)-presented left \(R\)-module \(N\). A right \(R\)-module \(V\) is called weakly \(n\)-flat, if \(\text{Tor}_n(V, N)=0\) for every \((n+1)\)-presented left \(R\)-module \(N\). In this paper, we give some characterizations and properties of \((n,d)\)-injective modules and \((n,d)\)-flat modules in the cases of \(n\geq d+1\) or \(n>d+1\). Using the concepts of weakly \(n\)-\(FP\)-injectivity and weakly \(n\)-flatness of modules, we give some new characterizations of left \(n\)-coherent rings.

MSC:

16D50 Injective modules, self-injective associative rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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