On regular power-substitution

Abstract

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular power-substitution if and only if a≂b in R implies that there exist n ∈ ℕ and a U ∈ GL _n(R) such that aU = Ub if and only if for any regular x ∈ R there exist m, n ∈ ℕ and U ∈ GL _n(R) such that x ^m I _n = x ^m Ux ^m, where a≂b means that there exists x, y, z ∈ R such that a = ybx, b = xaz and x = xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.

Keywords

Regular power-substitution / Regular power-cancellation / Stably free module

References

[1] Camps R., Menal P.. Power cancellation for artinian modules. Comm. Algebra, 1991, 19: 2081-2095

[2] Camps R., Menal P.. Power substitution property for rings of continuous functions. J. Algebra, 1993, 161: 455-466

[3] Chen H. Y.. Power-substitution, exchange rings and unit π-regularity. Comm. Algebra, 2000, 28: 5223-5233

[4] Chen H. Y.. Self-cancellation of modules having the finite exchange property. Chin. Ann. Math., 2005, 26B(1): 111-118

[5] Chen H. Y.. Separative ideals, clean elements and unit-regularity. Comm. Algebra, 2006, 34: 911-921

[6] Chen H. Y.. Exchange rings with stable range one. Czech. Math. J., 2007, 57(2): 579-590

[7] Chen H. Y.. On JB-rings. Chin. Ann. Math., 2007, 28B(6): 617-628

[8] Goodearl K. R.. Facchini A., Menini C.. von Neumann regular rings and direct sum decomposition problems, Abelian Groups and Modules. Math. Appl., 343, 1995, Dordrecht: Kluwer Academic Publishers 249-255

[9] Lam T. Y.. Modules with Isomorphic Multiples and Rings Isomorphic Matrix Rings, A Survey, 1999, Geneve: de l’Enseignement Mathematique

[10] Tuganbaev A. A.. Rings Close to Regular, 2002, Dordrecht: Kluwer Academic Publishers

[11] Wei J.. Power-substitution and exchange rings, Advances in Ring Theory, 2005, Hackensack: World Scientific Publishing 251-262

[12] Wu T.. The power-substitution condition of endomorphism rings of quasi-projective modules. Comm. Algebra, 2000, 28(1): 407-418