Bi-amalgamations subject to the arithmetical property

Abstract

Communicated by R. WiegandThis paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.

Bi-Amalgamation of Small Weak Global Dimension

“…In this paper, we characterize the bi-amalgamations of small weak global dimension. All obtained results recover and compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations ( [1,4,14,18]).…”

“…Note also that R is Noetherian since f (A) + J = Z/nZ is Noetherian ([13, Propo- Recall that a ring R is arithmetical if every finitely generated ideal is locally principal [9,12]. In [14], the authors proved that if w.dim(f (A) + J) ≤ 1, w.dim(g(A) + J ) ≤ 1, J ∩ Nil(B) = (0), J ∩ Nil(C) = (0), and for each m ∈ Max(A, I), J f (m)+J = (0) or J g(m)+J = (0), then w.dim A f,g (J, J ) ≤ 1. The converse holds if I is radical.…”

Some commutative ring extensions defined by almost Bézout condition

“…All their results recover known results on duplications and amalgamations. Recently in [16], the authors established necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property.…”

Weakly prime ideals issued from an amalgamated algebra

“…Moreover, in [9], they pursued the investigation on the structure of the rings of the form A ◃▹ f J, with particular attention to the prime spectrum, to the chain properties and to the Krull dimension. Amalgamation rings have been studied extensively, often because of their usefulness in constructing new classes of examples of rings satisfying various properties (for instance see [2,7,12,13]).…”