Ring elements of stable range one (Q6541322)

A ring \(R\) is said to have \textit{stable range one} if whenever \(aR+bR=R\) for some \(a,b\in R\), then there exists some \(c\in R\) such that \(a+bc\) is a unit of \(R\). This concept was first introduced in work of H.\ Bass from 1964, in the context of K-theory. Subsequently, it has become an important part of ring theory, having connections to many other conditions and properties, in particular unit-regularity and cancellation properties.\N\NIt is common to take an important property of rings and (whenever possible) distill it into an elementwise condition that captures said property at a ``local'' level. For instance, the modern concept of unit-regular elements captures the idea of Ehrlich's unit-regular rings elementwise. Similarly, when proving results about exchange rings, it usually suffices to merely assume that some element is ``suitable'' in the sense of Nicholson. Likewise, in 2005 the same two authors of the paper under review introduced an elementwise notion of stable range one; namely, an element \(a\in R\) is said to be of \textit{stable range one} if for any \(b\in R\) such that \(aR+bR=R\), then there exists some \(c\in R\) such that \(a+bc\) is a unit of \(R\). Since then, multiple authors have investigated this concept, and have generalized many theorems from the ring-theoretic setting to the elementwise case.\N\NThe observant reader may wonder why this is not called the \textit{right} stable range one condition, since it involves the \textit{right} unimodular equality \(aR+bR=R\). On the level of rings this is because Vaserstein showed that the definition is ultimately left-right symmetric. The symmetry question on the elementwise level remained open for nearly twenty years, with only a few special cases known, until the paper under review resolved this problem completely, in the positive. The proof proceeds by first showing that Jacobson's lemma\N\[\N\text{for all } a,b\in R,\ 1-ab\in U(R)\text{ implies } 1-ba\in U(R)\N\]\Ngeneralizes to a ternary ``super'' Jacobson's lemma\N\[\N\text{for all } a,b,x\in R,\ a+b-axb\in U(R)\text{ implies } a+b-bxa\in U(R),\N\]\Nand that this is exactly what is needed for the left-right symmetry of the elementwise stable range one property. (This ternary version of Jacobson's lemma has also appeared in a 1984 paper of \textit{P. Menal} and \textit{J. Moncasi} [J. Pure Appl. Algebra 33, 295--312 (1984; Zbl 0541.16021)].) Generalizing further, the authors show that the unit group \(U(R)\) in that ``super'' lemma can be replaced by the set of regular elements, and also by the set of unit-regular elements, and even by the set of stable range one elements.\N\NPleasantly, the paper develops the theory of stable range one elements from first principles, making this paper highly accessible. Important results from the literature are often given new conceptual proofs and generalized. Particular care is given to classifying those elements of stable range one inside natural classes and inside well-known rings. For example, those (von Neumann) regular elements with stable range one are given many different characterizations.\N\NAs another example, the authors fully classify those matrices in \(\mathbb{M}_n(\mathbb{Z})\) with stable range one; they are exactly the matrices whose determinant is in \(\{0,1,-1\}\). A number of other interesting results regarding matrices are proved, such as the fact that any matrix with just a single nonzero row, and with at least one zero entry on that row, always has stable range one. Underlying many of these results is their ``suspension'' theorem, stating that given any idempotent \(e\in R\), then \(a\in eRe\) has stable range one in the corner ring \(eRe\) if and only if \(a+(1-e)\) has stable range one in the ring \(R\).