Commutative rings in which every finitely generated ideal is quasi-projective
The main purposes of the paper are to define the notion of an fqp-ring and to investigate the correlation between the fqp-property and Prüfer conditions. Let \(R\) be a commutative ring. Let \(M\) be an \(R\)-module. An \(R\)-module \(V\) is said to be \(M\)-projective if the map \(\text{Hom}_R(V,M) \rightarrow \text{Hom}_R(V,M/N)\) is surjective for any submodule \(N\) of \(M\). It is said that \(V\) is quasi-projective if it is \(V\)-projective. An fqp-ring is defined as a ring in which every finitely generated ideal is quasi-projective. In section\,2, it is shown that a finitely generated \(R\)-module is quasi-projective if and only if it is a direct summand of \((R/I)^n\) for some ideal \(I\) of \(R\) and integer \(n\geq0\). In section\,3, the following theorems are given. (Theorem\,3.2) We have the following implications: \(R\) is an arithmetical ring \(\Rightarrow\) \(R\) is an fqp-ring \(\Rightarrow\) \(R\) is a Gaussian ring. And the implications are irreversible in general. (Theorem\,3.11) The weak global dimension of an fqp-ring is equal to \(0,1\) or \(\infty\). (Theorem\,3.14) Let \(R\) be a local ring. Then \(R\) is an fqp-ring if and only if \(R\) is a Prüfer ring and the total quotient ring of \(R\) is an fqp-ring. In section\,4, the fqp-property in trivial ring extensions is studied.
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