On Weakly Locally Finite Division Rings
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On Weakly Locally Finite Division Rings Trinh Thanh Deo1 · Mai Hoang Bien1 · Bui Xuan Hai1 Received: 15 May 2017 / Revised: 6 June 2018 / Accepted: 10 June 2018 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract Weakly locally finite division rings were considered in Deo et al. (J. Algebra 365, 42–49, 2012), where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer n ≥ 0 or n = ∞, we construct a weakly locally finite division ring whose Gelfand-Kirillov dimension is n. This fact shows in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Further, for the class of weakly locally finite division rings, we investigate some questions related with the well-known Kurosh Problem and with one of Herstein’s conjectures. Keywords Division rings · Weakly locally finite · Gelfand-Kirrilov dimension · Linear groups Mathematics Subject Classification (2010) 16K40 · 16P90
1 Introduction In the theory of algebras, it is well-known that an algebra is locally finite if its GelfandKirillov dimension (GKdim for short) is 0 [12]. Recall that the class of weakly locally finite division rings considered in [7] is a natural generalization of the class of locally finite division rings. As we will see in the text, a division ring is weakly locally finite if and
Trinh Thanh Deo
[email protected] Mai Hoang Bien [email protected] Bui Xuan Hai [email protected] 1
Faculty of Mathematics and Computer Science, VNUHCM - University of Science, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam
T. T. Deo et al.
only if it is locally PI. In 1996, Zhang [23, Example 5.7] gave an example of a locally PI division ring whose GKdim is 2. Therefore, the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, using Zhang’s idea, we construct the example of a locally PI division ring with GKdim = n ≥ 1 or ∞. Here, we use directly Mal’cev Neumann’s construction of the division ring of the free abelian group G of countable rank over some suitable base field K with respect to a certain group morphism Φ : G → Aut(K). Hence, we show in particular that there exist infinitely many weakly locally finite division rings that are not locally finite. Some readers of this manuscript called our attention to an old but unpublished example of J. C. McConnell on division ring with arbitrary Gelfand-Kirillov dimension (cf. [17]). From our discussion with J. C. McConnell and his colleagues, we felt that it is worth to have more examples on division rings of arbitrary predescribed Gelfand-Kirillov dimension. Further, we study some questions related with the Kurosh Problem for division rings. Recall that in 1941, Kurosh [13, Problem R] asked if a finitely generated algebraic algebra is necessarily a finite dimensional vector space over a base fi
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