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\((n,d)\)-cocoherent rings, \((n,d)\)-cosemihereditary rings and \((n,d)\)-\(V\)-rings. (English) Zbl 1486.16004

The author, in his earlier paper [Bull. Iran. Math. Soc. 40, No. 4, 809–822 (2014; Zbl 1338.16004)] had introduced the notions of \(n\)-cocoherent ring, \(n\)-cosemihereditary ring and \(n\)-\(V\)-ring. In this paper he generalizes these by defining the notions of \((n,d)\)-cocoherent ring, \((n,d)\)-cosemihereditary ring and \((n,d)\)-\(V\)-ring respectively. The author, in the same paper referred to above, had introduced the notion of \((n,d)\)-projective module. In this paper, he has generalized the notion of \((n,0)\)-projective module by defining a \((n,d)^*\)-projective module. The author derives some equivalent conditions for an \((n,d)\)-cosemihereditary ring and for an \((n,d)\)-\(V\)-ring. By defining the notion of \((n,d)\)-copure short exact sequence of \(R\)-modules, the author has derived some equivalent conditions for an \((n,d)^*\)-projective module in terms of \((n,d)\)-copure short exact sequences of \(R\)-modules. The author also introduces the notion of \((n,d)^*\)-projective dimension of a right \(R\)-module and gets equivalent conditions for \((n,d)^*\)-\(pd(M_R)\leq k\) for a right \(R\)-module \(M_R\) similar to equivalent conditions for \(pd(M_R)\leq k\) for a right \(R\)-module \(M_R\). The author concludes the paper by proving that over a commutative ring \(R\), every \((n,0)\)-projective \(R\)-module is \((n,0)\)-flat.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras
16E10 Homological dimension in associative algebras
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.

Citations:

Zbl 1338.16004
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References:

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