arXiv subjects
Publications and source records attributed to Samir Bouchiba.
We investigate the transfer of regularity between commutative, noetherian, local rings through a class of local homomorphisms which we call basically regular. We give numerical characterizations of these maps, investigate their behavior under composition and decomposition, and compare them with Avramov-Foxby-Herzog's weakly regular local homomorphisms.
This paper introduces and studies homological properties of new classes of modules, namely, the $\mathcal F_1$-flat modules and the $\mathcal F_1^{\fp}$-flat modules, where $\mathcal F_1$ stands for the class of right modules of flat dimension at most one and $\mathcal F_1^{\fp}$ its subclass consisting of finitely presented elements. This leads us to introduce a new class of rings that we term $\mathcal F_1^{\fp}$-coherent rings as they behave nicely with respect to $\mathcal F_1^{\fp}$-flat modules as do coherent rings with respect to flat modules. The new class of $\mathcal F_1^{\fp}$-coherent rings turns out to be a large one and it includes coherent rings, perfect rings, semi-hereditary rings and all rings $R$ such that $\lim\limits_{\lr}\mathcal P_1=\mathcal F_1$. As a particular case of rings satisfying $\lim\limits_{\lr}\mathcal P_1=\mathcal F_1$ figures the important class of integral domains.
Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring $R$. Actually, we translate the theory initiated by A. Grothendieck and R. Sharp and subsequently developed by A. Wadsworth on Krull dimension of tensor products of algebras over a field $k$ into the general setting of algebras over an arbitrary ring $R$. For this sake, we introduce and study the notion of a fibred AF-ring over a ring $R$. This concept extends naturally the notion of AF-ring over a field introduced by A. Wadsworth in \cite{W} to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fibre rings of tensor products of algebras over a ring. Also, given a triplet of rings $(R,A,B)$ consisting of two $R$-algebras $A$ and $B$ such that $A\otimes_RB\neq \{0\}$, we introduce the inherent notion to $(R,A,B)$ of a $B$-fibred AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product $A\otimes_RB$. As an application, we provide a formula for the Krull dimension of $A\otimes_RB$ when $A$ and $B$ are $R$-algebras with $A$ is zero-dimensional as well as for the Krull dimension of $A\otimes_{\mathbb{Z}}B$ when $A$ is a fibred AF-ring over the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an arbitrary ring. This enables us to answer a question of Jorge Matinez on evaluating the Krull dimension of $A\otimes_{\mathbb{Z}}B$ when $A$ is a Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that $A\otimes_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then dim$(A\otimes_{\mathbb{Z}}B)=\mbox {dim}\Big (\displaystyle {\frac B{2B}}\Big )$.
In this paper, we are concerned with the study of the dimension theory of tensor products of algebras over a field $k$. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any $k$-algebra $A$ such that the polynomial ring in one variable $A[X]$ is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of $A\otimes_kB$ for any $k$-algebra $A$ such that $A[X]$ is an AF-domain and any $k$-algebra $B$ generalizing the main theorem of Wadsworth in [16].
This paper is concerned with the study of the dimension theory of tensor products of algebras over a field $k$. We answer an open problem set in [6] and compute dim$(A\otimes_kB)$ when $A$ is a $k$-algebra arising from a specific pullback construction involving AF-domains and $B$ is an arbitrary $k$-algebra. On the other hand, we deal with the question (Q) set in [5] and show, in particular, that such a pullback $A$ is in fact a generalized AF-domain.