On pm$^+$ and finite character bi-amalgamation

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎Faculty‎ ‎of Science‎, ‎Box 11201 Zitoune‎, ‎University Moulay Ismail Meknes‎, ‎Morocco.

Abstract

‎Let $f:A\rightarrow B$ and $g:A \rightarrow C$ be two ring homomorphisms and let $J$ and $J^{'}$ be two ideals of $B$ and $C$‎, ‎respectively‎, ‎such that $f^{-1}(J)=g^{-1}(J^{'})$‎. ‎The bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{'})$ with respect of $(f,g)$ is the subring of $B\times C$ given by‎ ‎$A\bowtie^{f,g}(J,J^{'})=\{(f(a)+j,g(a)+j^{'})‎/ ‎a \in A‎, ‎(j,j^{'}) \in J\times J^{'}\}.$‎ ‎In this paper‎, ‎we study the transference of $pm^{+}$‎, ‎$pm$ and finite character ring-properties in the bi-amalgamation.

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D.D. Anderson and V.P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra 30 (2002), no. 7, 3327--3336.

W.D. Burgess and R. Raphael, On commutative clean rings and pm rings, in: Rings, Modules and Representations, pp. 35--55, Contemp. Math. 480, Amer. Math. Soc. Providence, RI, 2009.

M. Chiti and N. Mahdou, Some homological properties of an amalgamated duplication of a ring along an ideal, Bull. Iranian Math. Soc. 38 (2012), no. 2, 507--515.

M. Contessa, On pm-rings, Comm. Algebra 10 (1982), no. 1, 93--108.

M. Contessa, On certain classes of pm-rings, Comm. Algebra 12 (1984), no. 11-12, 1447--1469.

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, pp. 155--172, Walter de Gruyter, Berlin, 2009.

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 (2010), no. 9, 1633--1641.

M. D'Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2007), no. 2, 241--252.

M. D'Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (2007), no. 3, 443--459.

G. De Marco and A. Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc. 30 (1971) 459--466.

L. Gillman and M. Jerison, Rings of Continuous Functions, Grad. Texts in Math. 43, Springer-Verlag, Berlin, 1976.

M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1--27.

S. Kabbaj, K. Louartiti and M. Tamekkante, Bi-amalgamation algebras along ideals, J. Commut. Algebra, to appear.

M.D. Larsen, Prufer rings of finite character, J. Reine Angew. Math. 247 (1971) 92--96.

E. Matlis, Cotorsion Modules, Memoirs Amer. Math. Soc. 49 (1964), 66 pages.

E. Matlis, Decomposable modules, Trans. Amer. Math. Soc. 125 (1966) 147--179.

C.J. Mulvey, A generalization of Gelfand duality, J. Algebra 56 (1979) 499--505.

W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977) 278--279.

B. Olberding, Characterizations and constructions of h-local domains, in: Models, Modules and Abelian Groups, pp. 385--406, Walter de Gruyter, Berlin, 2008.