Najib Ouled AZAİEZ, Moutu ABDOU SALAM MOUTUİ

Some commutative ring extensions defined by almost Bézout condition

In this paper, we study the almost Bézout property in different commutative ring extensions, namely, in bi-amalgamated algebras and pairs of rings. In Section 2, we deal with almost Bézout domains issued from bi-amalgamations. Our results capitalize well known results on amalgamations and pullbacks as well as generate new original class of rings satisfying this property. Section 3 investigates pairs of rings where all intermediate rings are almost Bézout domains. As an application of our results, we characterize pairs of rings $(R,T)$, where $R$ arises from a $(T,M,D)$ construction to be an almost Bézout domain.

Keywords:

Bi-amalgamated algebra, almost Bézout domain pairs of rings, pullbacks,

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[1] M. Alqamoun and M. El Ouarrachi, Bi-amalgamations of rings defined by Bézout-like conditions, Palest. J. Math. 7 (2018), no. 2, 432-439.
[2] D. D. Anderson and M. Zafrullah, Almost Bezout domains, J. Algebra 142 (2), 285- 309, 1991.
[3] D. D. Anderson and M. Zafrullah, Almost Bezout domains, III, Bull. Math. Soc. Sci. Math. Roumanie, Tome 51, 3-9, 2008.
[4] D. D. Anderson, K. R. Knopp and R. L. Lewin, Almost Bezout domains, II, J. Algebra, 167, 547-556, 1994.
[5] A. Ayache, Maximal non-treed subring of its quotient field, Ric. Mat. 64 (1), 229-239, 2015.
[6] S. Bazzoni and S. Glaz, Gaussian properties of total rings of quotients, J. Algebra 310, 180-193, 2007.
[7] M. B. Boisen and P. B. Sheldon, CPI-extension: overrings of integral domains with special prime spectrum, Canad. J. Math. 29, 722–737, 1977.
[8] J.P. Cahen, Couple d’anneaux partageant un idéal, Arch. Math. 51, 505-514, 1988.
[9] M. D’Anna, A construction of Gorenstein rings, J. Algebra 306 (6), 507–519, 2006.
[10] M. D’Anna, C. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, in: M. Fontana, S. Kabbaj, B. Olberding, I. Swanson (Eds.), Commutative Algebra and its Applications, Walter de Gruyter, Berlin, 155-172, 2009.
[11] M. D’Anna, C. Finocchiaro and M. Fontana, Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (9), 1633-1641, 2010.
[12] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443-459, 2007.
[13] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
[14] A. Jaballah, Maximal non-Prüfer and Maximal non integrally closed subrings of a field, J. Algebra Appl. 11 (5), article ID: 1250041, 18pp, 2012.
[15] S. Kabbaj, K. Louartiti, and M. Tamekkante, Bi-amalgamated algebras along ideals, J. Commut. Algebra, 9, (1), 65-87, 2017.
[16] S. Kabbaj, N. Mahdou and M. A. S. Moutui, Bi-amalgamations subject to the arithmetical property, J. Algebra Appl., 16, 1750030 (11 pages), 2017.
[17] N. Mahdou, A. Mimouni and M. A. S. Moutui, On almost valuation and almost Bezout rings, Comm. Algebra 43, no. 1, 297–308, 2015.
[18] A. Mimouni, Prüfer-like conditions and pullbacks., J. Algebra 279 (2), 685–693, 2004.
[19] M. A. S. Moutui and N. Ouled Azaiez, Almost valuation property in bi–amalgamation and pairs of rings, J. Algebra Appl., published online, 2018, DOI:10.1142/S0219498819501044.
[20] A. R. Wadsworth, Pairs of domains where all intermediate domains are Noetherian, Trans. Amer. Math. Soc. 195, 201-211, 1974.