Commutative rings whose finitely generated ideals are quasi-flat

Abstract

A definition of quasi-flat left module is proposed and it is shown that any left module which is either quasi-projective or flat is quasi-flat. A characterization of local commutative rings for which each ideal is quasi-flat (resp. quasi-projective) is given. It is also proven that each commutative ring R whose finitely generated ideals are quasi-flat is of $λ$-dimension $\le$ 3, and this dimension $\le$ 2 if R is local. This extends a former result about the class of arithmetical rings. Moreover, if R has a unique minimal prime ideal then its finitely generated ideals are quasi-projective if they are quasi-flat. In [1] Abuhlail, Jarrar and Kabbaj studied the class of commutative fqp-rings (finitely generated ideals are quasi-projective). They proved that this class of rings strictly contains the one of arithmetical rings and is strictly contained in the one of Gaussian rings. It is also shown that the property for a commutative ring to be fqp is preserved by localization. It is known that a commutative ring R is arithmetical (resp. Gaussian) if and only if R M is arithmetical (resp. Gaussian) for each maximal ideal M of R. But an example given in [6] shows that a commutative ring which is a locally fqp-ring is not necessarily a fqp-ring. So, in this cited paper the class of fqf-rings is introduced. Each local commutative fqf-ring is a fqp-ring, and a commutative ring is fqf if and only if it is locally fqf. These fqf-rings are defined in [6] without a definition of quasi-flat modules. Here we propose a definition of these modules and another definition of fqf-ring which is equivalent to the one given in [6]. We also introduce the module property of self-flatness. Each quasi-flat module is self-flat but we do not know if the converse holds. On the other hand, each flat module is quasi-flat and any finitely generated module is quasi-flat if and only if it is flat modulo its annihilator. In Section 2 we give a complete characterization of local commutative rings for which each ideal is self-flat. These rings R are fqp and their nilradical N is the subset of zerodivisors of R. In the case where R is not a chain ring for which N = N 2 and R N is not coherent every ideal is flat modulo its annihilator. Then in Section 3 we deduce that any ideal of a chain ring (valuation ring) R is quasi-projective if and only if it is almost maximal and each zerodivisor is nilpotent. This complete the results obtained by Hermann in [11] on valuation domains. In Section 4 we show that each commutative fqf-ring is of $λ$-dimension $\le$ 3. This extends the result about arithmetical rings obtained in [4]. Moreover it is shown that this $λ$-dimension is $\le$ 2 in the local case. But an example of a local Gaussian ring R of $λ$-dimension $\ge$ 3 is given.