Gaussian Trivial Ring Extensions and FQP-rings

Abstract

Abstract. Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqpring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1 or ∞.Trivial ring extensions are often used to give either examples or counterexamples of rings. One of the most famous is the chain ring R which is not factor of a valuation domain (see [8, X.6] and [7, Theorem 3.5]). This ring R is the trivial ring extension of a valuation domain D by a non-standard uniserial divisible D-module. This example gives a negative answer to a question posed by Kaplansky.In [14], [15] and [1] there are many results on trivial ring extensions and many examples of such rings. In particular, necessary and sufficient conditions are given for the trivial ring extension of a ring A by an A-module E to be either arithmetical or Gaussian in the following cases: either A is a domain and K is its quotient field, or A is local and K is its residue field, and E is a K-vector space.In our paper more general results are shown. For instance the trivial ring extension R of a ring A by a non-zero A-module E is a chain ring if and only if A is a valuation domain and E a divisible module, and R is Gaussian if and only if A is Gaussian and E verifies aE = a 2 E for each a ∈ A. Complete characterizations of arithmetical trivial ring extensions are given too. But Bezout trivial ring extensions of a ring A are characterized only in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected.We also study trivial ring extensions which are fqp-rings. The class of fqprings was introduced in [1] by Abuhlail, Jarrar and Kabbaj. We get a complete characterization of trivial ring extensions which are fqp-ring only in the local case. Each fqp-ring is locally fqp and the converse holds if it is coherent, but this is not generally true. We introduce the class of fqf-rings which satisfy the condition "each finitely generated ideal is flat modulo its annihilator", and it is exactly the class of locally fqp-rings. So, trivial fqf-ring extensions are completely characterized. This new class of rings contains strictly the class of fqp-rings.In [6] it is proven that each arithmetical ring has a finitistic weak dimension equal to 0, 1 or 2. We show that any fqf-ring satisfy this property too, and its global weak dimansion is 0, 1 or ∞ as it is shown for fqp-rings in [1].2010 Mathematics Subject Classification. 13F30, 13C11, 13E05.

Bi-amalgamations subject to the arithmetical property

“…where w. dim(R) denotes the weak global dimension of R. Recall that all these properties are identical to the notion of Prüfer domain if R has no zero-divisors, and that the above implications are irreversible, in general, as shown by examples provided in [1,4,5,6,8,9,10,12,13,22,23,24]. Very recently, these conditions (among other Prüfer conditions) were thoroughly investigated in various contexts of duplications [12].…”

Zaks' conjecture on rings with semi-regular proper homomorphic images

“…A ring R is an fqp-ring if every finitely generated ideal of R is quasi-projective [1,11]. We always have:…”

“…Let R be a ring and [1,11]. We always have: Arithmetical ⇒ fqp ⇒ Gaussian and the fqp notion is a local property in the class of coherent rings [11,Proposition 4.4] or [1,Corollary 3.15]. Lemma 2.6.…”

Commutative rings whose finitely generated ideals are quasi-flat

“…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].…”