The Kaplansky condition and rings of almost stable range 1 (Q2845450)

All rings in this review are commutative with unity. Two rectangular matrices \(A\) and \(B\) in \(M_{s,t}(R)\) are equivalent if there exist invertible matrices \(P\in M_{s,s}(R)\) and \(Q\in M_{t,t}(R)\) such that \(B=PAQ\). The ring \(R\) is \(K\)-Hermite if every rectangular matrix \(A\) over \(R\) is equivalent to an upper or a lower triangular matrix (the term `\(K\)-Hermite ring' was introduced by \textit{T. Y. Lam} [Serre's problem on projective modules. Springer Monographs in Mathematics. Berlin: Springer Verlag, (2006; Zbl 1101.13001)]). A ring \(R\) is an elementary divisor ring (EDR) if and only if every rectangular matrix \(A\) over \(R\) is equivalent to a diagonal matrix. By Irving Kaplansky, a \(K\)-Hermite ring \(R\) is an EDD ring if and only if it satisfies condition: for any three elements \(a,b,c\) in \(R\) that generate the ideal \(R\), there exist elements \(p,q\in R\) so that \((pa,pb+qc)=R\) (the Kaplansky's condition) [\textit{I. Kaplansky}, Trans. Am. Math. Soc. 66, No. 2, 464--491 (1949; Zbl 0036.01903)].NEWLINENEWLINEThe author presents some variants of the Kaplansky condition. One of the results:NEWLINENEWLINETheorem 2.9. Let \(R\) be a \(K\)-Hermite ring. The following conditions are equivalent:NEWLINENEWLINE(1) \(R\) is an elementary divisor ring.NEWLINENEWLINE(2) For any elements \(x,y,z\in R\) such that \((x,y)=R\), there exists an element \(\lambda\in R\) such that \(x+\lambda y=uv\), where \((u,z)=(v,1-z)=R\). Moreover, the elements \(u\) and \(v\) can be chosen such that \((u,v)=R\).NEWLINENEWLINEA ring \(R\) is called Bézout ring if each finitely generated ideal of \(R\) is principal. The author presents an example of a Bézout ring that is an elementary divisor ring but does not have almost stable rang 1. This result is the answer to a question of \textit{W. Wm. McGovern} [J. Pure Appl. Algebra 212, No 2, 340--348 (2008, Zbl 1159.13010)].