Unitriangular groups and undecidability (Q1901892)

For an associative ring with unit \(R\), we denote by \(\text{UT}_n (R)\) the group of all upper unitriangular \((n \times n)\)-matrices over \(R\). Since the group \(\text{UT}_n (R)\), obviously, is interpretable in the ring \(R\) without parameters, \(\text{Th(UT}_n (R))\) reduces by Turing to \(\text{Th} (R)\); in particular, irreducibility of \(\text{Th(UT}_n (R))\) implies irreducibility of \(\text{Th} (R)\). \textit{A. I. Mal'tsev} [Mat. Sb., Nov. Ser. 50(92), 257-266 (1960; Zbl 0100.014)] proved that the ring \(R\) can be interpreted in the group \(\text{UT}_3 (R)\) with parameters; so, hereditary irreducibility of \(\text{Th}(R)\) implies hereditary irreducibility of \(\text{Th(UT}_n(R))\). In Sib. Mat. Zh. 33, No. 4, 24-29 (1992; Zbl 0781.03021), the author had shown that the irreducibility of \(\text{Th} (R)\), generally speaking, does not imply the irreducibility of \(\text{Th(UT}_3 (R))\). Moreover, for all Turing degrees \(d_1\) and \(d_2\) such that \(d_1 \leq d_2\), there exists a ring with unit \(R\) such that \(\text{Th(UT}_3 (R))\) has the degree \(d_1\), and \(\text{Th} (R)\) has the degree \(d_2\). But if \(R\) is either a skew field or a commutative ring, then \(\text{Th(UT}_3(R))\) and \(\text{Th} (R)\) are recursively equivalent. In this article we strengthen these results. Theorem 1. If \(R\) is a commutative or integral associative ring, then \(\text{Th} (R)\) and \(\text{Th(UT}_n(R))\) are recursively equivalent for all \(n \geq 3\). Theorem 2. For all Turing degrees \(d_1\) and \(d_2\) such that \(d_1 \leq d_2\) there exists an associative ring \(R\) such that for any \(n \geq 3\) \(\text{Th(UT}_n (R))\) has the degree \(d_1\), and \(\text{Th} (R)\) has the degree \(d_2\).