Nil skew α-Armendariz amalgamated rings

Subject Areas : Statistics

Negin Farshad ¹ (Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran)
Shaaban Ali Safarisabet ² (Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran)
Ahmad Moussavi ³ (Faculty of Mathematical Sciences, Department of Pure Mathematics, Tarbiat Modares University, Tehran, Iran)

Received: 2021-06-07 Accepted : 2021-09-13 Published : 2024-01-18

Keywords: -آرمنداریز اریب ضعیف, 2-اولیه, آرمنداریز اریب پوچ, &alpha, حلقه در هم آمیخته, -&alpha,

Abstract :

Let f :A→B be a ring homomorphism and K an ideal of B. In this paper we determine the endomorphism α of the amalgamation ring A⋈^f K of A and B along K with respect to f. Then we investigate some annihilation properties, such as nil skew α-Armendariz, and weak skew α-Armendariz, of the amalgamation ring A⋈^f K. We show the relations among A , f(A)+K and the amalgamation ring A⋈^f K, in terms of their nil skew Armendariz and weak skew Armendariz properties. Also we investigate 2-primal property of the amalgamation ring A⋈^f K. Among other results, we show that, if A is a 2-primal α_1-compatible ring and f(A) +K is a 2-primal α_2-compatible ring, then the amalgamation ring A⋈^f K is a nil skew α-Armendariz ring, where α_1 and α_2 are endomorphisms of A and f(A)+K, respectively, and α is the endomorphism on A⋈^f K induced by α_1 and α_2.

References:


[1] M. D’ Anna, C. A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, Commutative algebra and its applications, Walter de Gruyter, Berlin, (2009) 241-252.
[2] M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci, 73 (1997) 14-17.
[3] D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra, 26 (7) (1998) 2265-2272.
[4] Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring. J. Pure and Appl. Algebra, 168 (2002) 45-52.
[5] N. Mahdou, A. Mimouni, M. EL Ouarrachi, On Armendariz-like properties in amalgamated algebras along ideals, Turk. J. Math. 41 (2017) 1673-1686.
[6] C. Y. Hong, N. K. Kim, T. K. Kwak, On skew Armendariz rings, Comm. Algebra, 31 (1) (2003) 103-122.
[7] C. Zhang and J. Chen, Weak skew Armendariz rings, J. Korean Math. Soc., 47 (3) (2010) 455-466.
[8] M. Habibi and A. Moussavi, On nil skew Armendariz rings, Asian-European J. Math, 5 (2) (2012).
[9] G. F. Birkenmeier, H. E. Heatherly, E. K. Lee, Completely primre ideals and radicals in Near-rings. Proc. of Near-Rings and Near-Fields, edited by Y. Fong et al., Kluwer (1995) 63-73.
[10] G. F. Birkenmeier, H. E. Heatherly, E. K. Lee, Completely primre ideals and associated radicals, in (eds. S. K. Jain, S. T. Rizvi) Ring Theory (Granville, OH, 1992), World Scientific, Singapore and River Edge (1993), 102-129,373. MR 96e:16025.
[11] C. Huh, H. K. Kim, Y. Lee, Questions on 2-primal rings, Comm. Algebra, 26 (2) (1998) 595-600.
[12] G. Marks, Skew polynomial rings over 2-primal rings, Comm. Algebra, 27 (9) (1999) 4411-4423.
[13] Z. K. Liu and R. Y. Zhao, On weak Armendariz rings, Comm. Algebra, 34 (7) (2006) 2607-2616.
[14] Paul E. Bland, Rings and their modules, Walter de Gruyter GmbH and Co. KG, Berlin New York (2011).
‎[15] M. Habibi and A. Moussavi, On nil skew Armendariz rings, Asian-European J. Math., 5 (2) (2012).
[16] ‎‎ L. Ouyang and G. F. Birkenmeier, Weak annihilator over extension rings, Bull. Malays. Math. Sci. Soc., 35 (2) (2012) 345-357.
[17] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar., 107 (2005) 207-224.
[18] L. Ouyang, Extensions of generalized α-rigid rings, Inter. elec. j. alg., 3 (2008) 103-116.