On \(n\)-coherent rings and \((n,d)\)-rings. (English) Zbl 1089.16001

Let \(R\) be an associative ring and \(n,d\) be non-negative integers. A right module \(M\) is called \((n,d)\)-injective (resp. flat), if \(\text{Ext}^{d+1}(P,M)=0\) (resp. \(\text{Tor}_{d+1}(M,Q)=0\)) for every \(n\)-presented right (resp. left) module \(P\) (resp. \(Q\)). D. L. Costa [Commun. Algebra 22, No. 10, 3997–4011 (1994; Zbl 0814.13010)] introduced the concept of \(n\)-coherent rings. In this paper, the characterizations of coherent rings are extended to \(n\)-coherent rings using the preceding concepts of \((n,d)\)-injective and \((n,d)\)-flat. Next, the author studies \((n,d)\)-injective and \((n,d)\)-flat preenvelopes and precovers for \(n\)-coherent rings. In the last section, it is shown that if \(R\leq S\) is an almost excellent extension, \(R\) is a right (resp. weak) \((n,d)\)-ring if and only if \(S\) is a right (resp. weak) \((n,d)\)-ring.


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


Zbl 0814.13010
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