Bézout rings with almost stable range 1 (Q2457279)

Let \(R\) be a PID and let \(A\) be a matrix over \(R.\) Then \(A\) admits diagonal reduction, i.e., there exist invertible matrices \(P,Q\) such that \( PAQ=\text{diag}(d_{1},d_{2},\dots,d_{n},\dots).\) \textit{I. Kaplansky} [Trans. Am. Math. Soc. 66, 464--491 (1949; Zbl 0036.01903)] sought to study more general rings, with \(1\neq 0,\) that had the diagonal reduction property, calling them elementary divisor rings EDR's. He showed that to verify that a ring \(R\) is an elementary divisor ring it is sufficient to show that all the \(1\times 2,\) \(2\times 1\) and \( 2\times 2\) matrices admitted diagonalization. He called \(R\) a Hermite ring if all \(1\times 2\) and \(2\times 1\) matrices over \(R\) admit diagonal reduction. So, EDR = Hermite + X. It is easy to see that a commutative Hermite ring is Bézout (every \(2\)-generated ideal is principal). He showed that for the integral domain case X is provided by an adequate domain. Here a commutative ring \(R\) is adequate if for all pairs \(a,b\in R,\) we can split \(a\) as a product \(a=rs\) such that \(rR+bR=R\) and for every nonunit divisor \( s^{\prime }\) of \(s\) we have \(s^{\prime }R+bR\neq R.\) We shall henceforth mean by a ring \(R\) a commutative ring with 1. Most of the work on commutative EDR's essentially follows the template provided by Kaplansky. \textit{L. Gillman} and \textit{M. Henriksen} [Trans. Am. Math. Soc. 82, 362--365 (1956; Zbl 0073.02203)] studied commutative rings with zero divisors, showing that a commutative ring \(R\) is a Hermite ring if and only if for each pair \(a,b\) in \(R\) there are elements \( a_{1},b_{1},d\in R\) such that \(a=a_{1}d,b=b_{1}d\) and \(a_{1}R+b_{1}R=R.\) They also provide X, mentioned above, in a result that is reminiscent of Theorem 5.2 of Kaplansky's paper mentioned above. \textit{M. Henriksen} [Mich. Math. J. 3, 159--163 (1956; Zbl 0073.02301)] showed that X = for every pair \(a,b\in R,\) with \(a\notin J(R),\) there is \(r\in R\) such that \(V(r)=V(a)\backslash V(b)\) where \(V(a)\) denotes the set of all maximal ideals containing \(a\in R\); (this condition is called Henriksen's hypothesis in the paper under review). Henriksen also showed that a Hermite ring \(R\) is and EDR if and only if \( R/J(R)\) is an EDR. \textit{M. D. Larsen, W. J. Lewis} and \textit{T. S. Shores} [Trans. Am. Math. Soc. 187, 231--248 (1974; Zbl 0283.13002)] showed that X is provided by (N): For all \(a,b\in R\) with \(a\notin J(R)\), the Jacobson radical of \(R,\) (1) there exists \(m\in R\) such that \(bR+mR=R\) and (2) if there is \(n\in R\) with \(nR+aR\neq R,\) yet \( nR+bR=R\) then \(nR+mR\neq R.\)They also showed that Henriksen's hypothesis implies (N) and the author of the paper under review shows, in section 2, that for a Bézout ring \(R\) both (N) and Henriksen's hypothesis are equivalent. A ring \(R\) is said to have stable range 1 if for \(a,b\in R\), \(aR+bR=R\) implies that \(a+xb\) is a unit in \(R\), for some \(x\in R.\) Also, \(R\) has almost stable range 1 if every proper homomorphic image of \(R\) has stable range 1. Next, \(R\) has stable range 2 if for \(a,b,c\in R\) with \(aR+bR+cR=R\) implies \((a+cx)R+(b+cy)R=R\) for some \(x,y\in R.\) According to \textit{B. V. Zabav'skyi} [Ukr. Math. J. 55, No. 4, 665--670 (2003); translation from Ukr. Mat. Zh. 55, No. 4, 550--554 (2003; Zbl 1038.16008)] a Bézout ring \(R\) is a Hermite ring if and only if \(R\) has stable range 2. Using this result the author shows, in section 3, that a Bézout ring \(R\) is an EDR if and only if \(R/J(R)\) is and EDR a definite improvement on Henriksen's result that required \(R\) to be Hermite. As a further consequence he shows that for a ring \(R\) with almost stable range 1 being Bézout is equivalent to being an EDR. In this section the author also gives examples of rings with almost stable range 1. In the fourth and final section the author constructs, via Jaffard-Kaplansky-Ohm Theorem, an example of a Bézout domain that does not satisfy Henriksen's hypothesis. The paper is jam-packed with interesting results, but is highly readable.