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AMALGAMATED MODULES ALONG AN IDEAL
El Khalfaoui, Rachida,Mahdou, Najib,Sahandi, Parviz,Shirmohammadi, Nematollah Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.1
Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to , and defined by M ⋈ JN := {(m, (m) + n) | m ∈ M and n ∈ JN}, where : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.
The dimension of the maximal spectrum of some ring extensions
Rachida El Khalfaoui,Najib Mahdou 대한수학회 2023 대한수학회논문집 Vol.38 No.4
Let $A$ be a ring and $\J = \{\text{ideals $I$ of $A$} \,|\, J(I) = I\}$. The Krull dimension of $A$, written $\dim A$, is the sup of the lengths of chains of prime ideals of $A$; whereas the dimension of the maximal spectrum, denoted by $\dim_\J A$, is the sup of the lengths of chains of prime ideals from $\J$. Then $\dim_{\J} A\leq \dim A$. In this paper, we will study the dimension of the maximal spectrum of some constructions of rings and we will be interested in the transfer of the property $J$-Noetherian to ring extensions.