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Structure theory of p.p. rings and their generalizations. (English) Zbl 1477.14005

Summary: In this paper, new advances on the understanding the structure of p.p. rings and their generalizations have been made. Especially among them, it is proved that a commutative ring \(R\) is a generalized p.p. ring if and only if \(R\) is a generalized p.f. ring and its minimal spectrum is Zariski compact, or equivalently, \(R/\mathfrak{N}\) is a p.p. ring and \(R_{\mathfrak{m}}\) is a primary ring for all \(\mathfrak{m}\in\mathrm{Max}(R)\). Some of the major results of the literature either are improved or are proven by new methods. In particular, we give a new and quite elementary proof to the fact that a commutative ring \(R\) is a p.p. ring if and only if \(R[x]\) is a p.p. ring.

MSC:

14A05 Relevant commutative algebra
13A15 Ideals and multiplicative ideal theory in commutative rings
13C10 Projective and free modules and ideals in commutative rings
13C11 Injective and flat modules and ideals in commutative rings
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