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 $lambda$-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 $lambda$-dimension $le$ 3. This extends the result about arithmetical rings obtained in [4]. Moreover it is shown that this $lambda$-dimension is $le$ 2 in the local case. But an example of a local Gaussian ring R of $lambda$-dimension $ge$ 3 is given.
Let R be a commutative ring. If P is a maximal ideal of R whose a power is finitely generated then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension $le$ 2. And
if P is a prime ideal of R whose a power is finitely generated then we show that P is finitely generated if R is either a reduced coherent ring or a polynomial ring over a reduced arithmetical ring. These results extend a theorem of Roitman, published in 2001, on prime ideals of coherent integral domains.
Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for instance,
if R is left perfect but not left pure-semisimple then each left R-module is cotorsion but there exist non-pure-injective left modules. The aim of this paper is to describe the class C of commutative rings R for which each cotorsion R-module is pure-injective. It is easy to see that C contains the class of von Neumann regular rings and the one of pure-semisimple rings. We prove that C is strictly contained in the class of locally pure-semisimple rings. We state that a commutative ring R belongs to C if and only if R verifies one of the following conditions: (1) R is coherent and each pure-essential extension of R-modules is essential; (2) R is coherent and each RD-essential extension of R-modules is essential; (3) any R-module M is pure-injective if and only if Ext 1 R (R/A, M) = 0 for each pure ideal A of R (Baers criterion).
It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $widehat{R}$ of $R$ is of finite rank over the completion $widetilde{R}$ of $R$ in the $R$-topology. In t
his case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over one Krull-dimensional local Noetherian rings has finite Malcev rank. The preservation of Goldie dimension finiteness by localization is investigated too.
It is proven each ring $R$ for which every indecomposable right module is pure-projective is right pure-semisimple. Each commutative ring $R$ for which every indecomposable module is pure-injective is a clean ring and for each maximal ideal $P$, $R_P
$ is a maximal valuation ring. Complete discrete valuation domain of rank one are examples of non-artinian semi-perfect rings with pure-injective indecomposable modules.
It is proven that each commutative arithmetical ring $R$ has a finitistic weak dimension $leq 2$. More precisely, this dimension is 0 if $R$ is locally IF, 1 if $R$ is locally semicoherent and not IF, and 2 in the other cases.
In this paper, we compare $(n,m)$-purities for different pairs of positive integers $(n,m)$. When $R$ is a commutative ring, these purities are not equivalent if $R$ doesnt satisfy the following property: there exists a positive integer $p$ such that
, for each maximal ideal $P$, every finitely generated ideal of $R_P$ is $p$-generated. When this property holds, then the $(n,m)$-purity and the $(n,m)$-purity are equivalent if $m$ and $m$ are integers $geq np$. These results are obtained by a generalization of Warfields methods. There are also some interesting results when $R$ is a semiperfect strongly $pi$-regular ring. We also compare $(n,m)$-flatnesses and $(n,m)$-injectivities for different pairs of positive integers $(n,m)$. In particular, if $R$ is right perfect and right self $(aleph_0,1)$-injective, then each $(1,1)$-flat right $R$-module is projective. In several cases, for each positive integer $p$, all $(n,p)$-flatnesses are equivalent. But there are some examples where the $(1,p)$-flatness is not equivalent to the $(1,p+1)$-flatness.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in add
ition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prufer domain of finite character, localizations of injective $R$-modules are injective.
Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $mathbb{M}_n(R)$ is strongly clean for each integer $
n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $pi$-regular commutative ring.
