Journal of Advanced Mathematical Modeling

Journal of Advanced Mathematical Modeling

where Goodearl does not work

Document Type : Original Paper

Authors
Ehsan Momtahan 1
Afshin Amini 2
Babak Amini 2
1 Department of Mathematics, Faculty of Science, Yasuj University, Yasuj, Iran
2 Department of Mathematics, Shiraz University
10.22055/jamm.2025.48284.2328
Abstract
K. Goodearl has constructed a unit regular ring ovIn this article, we review Goodearl's example from the aspect of being Villamayer and check whether this example is really able to generate a regular and Villamayer ring on which there is a non-Arthenian Noetherian module. Among other results, we will show that Goodearl's example, despite its originality, is not up to the mark. We will also show that Goodearl's example for the first infinite ordinal is primitive from both side.
Keywords
"Regular rings
Villmayor ring
Semi-artinian ring
noetherian module"
Subjects
[1] Baccella, G.,1995. semi-artinian V-rings and semi-artinian von Neumann regular rings, J. Algebra 173, pp 587-612.
[2] Elliott, G A., 1976. On the Classification of Inductive Limits of Sequences of Semisimple FiniteDimensional Algebras J. Algebra, 38, pp 29-44.
[3] K.R.Goodearl, 1979. Von Neumann Regular Rings, Pitman, London.
[4] Gooderl, K.R., 1980. Artinian and Noetherian modules over regular rings. Comm. Algebra, 8(5), pp.
477-504. doi:10.1080/00927878008822470.
[5] Goursaud, J.M. and Jeremy. L. 1977. Une notion de rang dans les facteurs reguliers autoinjectifs a
droite, Comm. Alg. 5 , pp 829–839.
[6] Fisher, JW. and Snider, R., 1974. Prime von Neumann regular rings and primitive group rings.
Proc.Amer.Math.Soc, 44, (2) doi:10.2307/2040414
[7] Herbera, D., 2009. Simple modules over small rings, Modules and representations, Contemp.
Math.,480, Amer. Math. Soc., Providence, RI, pp 189-205.
[8] Momtahan, E., 2011. On Semi-Artinian weakly co-emisimple modules, Bultein of IMS,(37), 3, pp 149-157.
[9] Osofsky, B., 1966, cyclic injective modules of full linear rings, Proc. Amer. Math. Soc., 17, pp 247-252.
[10] Wisbuer, R., 1991. Foundations of Rings and Module Theory, Gordon and Breach

  • Receive Date 03 November 2024
  • Revise Date 23 April 2025
  • Accept Date 04 September 2025