A note on symmetric elements of division rings with involution (Q2141415)

Let \(R\) be a ring and \(R[t]\) the polynomial ring in an indeterminate \(t\) over \(R\), commuting with the elements of \(R\). Assume that \(\Lambda\) is a ring including \(R\) as a subring, and let \(\alpha\) be an arbitrary element of \(\Lambda\). For any nonzero polynomial \(f(t) = \sum_{i=0}^n a_it^i\), the left (respectively, right) evaluation of \(f\) at \(\alpha\) is defined to be the element \(f_{\ell}(\alpha ) = \sum_{i=0}^n \alpha^ia_i\) (respectively, \(f_r(\alpha ) = \sum_{i=0}^n a_i\alpha^i\)); also, by definition, the left and the right evaluation at \(\alpha\) of the zero polynomial are equal to the zero element of \(R\). We say that \(\alpha\) is a left root of \(f\) if \(f_{\ell}(\alpha ) = 0\); when \(f_r(\alpha ) = 0\), \(\alpha\) is called a right root of \(f\). If \(\alpha\) is a left root (resp. right root) of some nonzero polynomial in \(R[t]\), then it is said to be right algebraic (resp. left algebraic) over \(R\). The two notions coincide in case \(R\) is included in the centre \(Z(\Lambda )\) of \(\Lambda\). A subset \(A\) of \(\Lambda\) is called right algebraic (resp. left algebraic) over \(R\) if \(A\) consists of right algebraic (resp. left algebraic) elements over \(R\). One-sided algebraicity has been introduced to study the roots of polynomials over division rings (see [\textit{N. Jacobson}, Structure of rings. Revised edition. Providence, RI: AMS (1964); \textit{T. Y. Lam}, A first course in noncommutative rings. New York etc.: Springer-Verlag (1991; Zbl 0728.16001)]). Under the hypothesis that \(R\) is a field and \(\Lambda\) is a division ring, it is presently unknown whether \(\Lambda\) is left algebraic over \(R\) if and only if it is right algebraic over \(R\). When \(\Lambda\) and \(R\) are noncommutative division rings, the answer to the analogous question is, generally, negative [\textit{B. X. Hai} et al., Arch. Math. 113, No. 2, 141--148 (2019; Zbl 1433.16019)]. The paper under review provides versions of several known results on one-sided algebraicity, including the above-noted one [\textit{J. P. Bell} and \textit{D. Rogalski}, Math. Z. 277, No. 1--2, 591--609 (2014; Zbl 1311.16013); \textit{J. P. Bell} et al., J. Pure Appl. Algebra 217, No. 9, 1605--1610 (2013; Zbl 1293.16016)], and Corollary~3 of [\textit{M. A. Chebotar} et al., Manuscr. Math. 113, No. 2, 153--164 (2004; Zbl 1054.16012)] for division rings with involution. As usual, by an involution on a ring \(R\), we mean a map \({\ast}: R \to R\), \(x \to x^{\ast}\), such that \((a^{\ast})^{\ast} = a\), \((a + b)^{\ast} = a^{\ast} + b^{\ast}\) and \((ab)^{\ast} = b^{\ast}a^{\ast}\), for all \(a, b \in R\). An element \(x \in R\) is said to be symmetric if \(x^{\ast} = x\). Henceforth, we assume that \(D\) is a division ring with involution \({\ast}\), \(F\) is the centre of \(D\), \(S\) is the set of symmetric elements of \(D\), and \(K\) is a division subring of \(D\) which includes \(F\) as a subset. The first main result of the paper under review states that if the field \(F\) is uncountable, then \(D\) is right algebraic over \(K\) if and only if so is \(S\). This result allows the author to construct an example of division rings \(K \subset D\), such that \(D\) is right algebraic but not left algebraic over \(K\).