##
**Rings related to stable range conditions.**
*(English)*
Zbl 1245.16002

Series in Algebra 11. Hackensack, NJ: World Scientific (ISBN 978-981-4329-71-2/hbk; 978-981-4329-72-9/ebook). xviii, 661 p. (2011).

This book is devoted to the study of conditions on rings and modules related to the classical stability conditions of K-theory. Characterizations, equivalent forms, and consequences are developed, among which cancellation for direct sums of modules and diagonal reduction of matrices are recurring themes. Much of the development is carried out for the class of exchange rings introduced by R. B. Warfield, jun. [Math. Ann. 199, 31-36 (1972; Zbl 0228.16012)], a large class that contains, for instance, all von Neumann regular rings, \(\pi\)-regular rings, semiperfect rings, semi-Artinian rings, clean rings, and \(C^*\)-algebras of real rank zero.

The fundamental results in which cancellation is obtained entirely from stable range conditions were established by E. G. Evans, jun. [Pac. J. Math. 46, 115-121 (1973; Zbl 0272.13006)] and R. B. Warfield, jun. [Pac. J. Math. 91, 457-485 (1980; Zbl 0484.16017)]. Evans’ result states that if the endomorphism ring of a module \(A\) has stable range \(1\), then \(A\) cancels from direct sums: \(A\oplus B\cong A\oplus C\) implies \(B\cong C\). Warfield proved that if the endomorphism ring of \(A\) has stable range \(\leq n\) and \(B\) has a direct summand isomorphic to \(\bigoplus^nA\), then \(A\oplus B\cong A\oplus C\) implies \(B\cong C\). In the case of a von Neumann regular ring \(R\), L. Fuchs [Monatsh. Math. 75, 198-204 (1971; Zbl 0225.16013)] and I. Kaplansky [“Bass’s first stable range condition”, Mimeographed notes (1971)] (results incorporated in [L. N. Vaserstein, J. Pure Appl. Algebra 34, 319-330 (1984; Zbl 0547.16017)]) proved that \(R\) has stable range \(1\) if and only if it has internal cancellation: \(R=A_1\oplus B_1=A_2\oplus B_2\) (as modules) with \(A_1\cong A_2\) implies \(B_1\cong B_2\); this was extended to cancellation for finitely generated projective \(R\)-modules by D. Handelman [J. Algebra 48, 1-16 (1977; Zbl 0363.16009)]. Later, H.-P. Yu proved that an exchange ring \(R\) has stable range \(1\) if and only if the finitely generated projective \(R\)-modules have cancellation [J. Pure Appl. Algebra 98, No. 1, 105-109 (1995; Zbl 0837.16009)].

Diagonal reduction of matrices \(X\) (the existence of invertible matrices \(U\) and \(V\) such that \(UXV\) is diagonal) was first related to cancellation conditions in the case of a von Neumann regular ring \(R\), by P. Menal and J. Moncasi [J. Pure Appl. Algebra 24, 25-40 (1982; Zbl 0484.16006)], who proved that all (rectangular) matrices over \(R\) admit diagonal reduction if and only if \(R^2\oplus B\cong R\oplus C\) implies \(R\oplus B\cong C\) for all \(R\)-modules \(B\), \(C\). Ara, O’Meara, Pardo and the reviewer proved that over an exchange ring \(R\), all von Neumann regular square matrices admit diagonal reduction if and only if the finitely generated projective \(R\)-modules satisfy separative cancellation: \(A\oplus A\cong A\oplus B\cong B\oplus B\) implies \(A\cong B\) [P. Ara, K. R. Goodearl, K. C. O’Meara and E. Pardo, Linear Algebra Appl. 265, 147-163 (1997; Zbl 0883.15006)].

The classical stable range \(1\) condition for a unital ring \(R\) can be stated as follows: If \(a,b\in R\) and \(aR+bR=R\), there exists \(y\in R\) such that \(a+by\) is a unit. (It is known that this condition is left-right symmetric.) Many variations are investigated in the book, where “\(a+by\) is a unit” is replaced by “\(a+by\) has property (U)” and sometimes a restriction is imposed on \(y\). Here are some samples: (1) unit \(1\)-stable range: \(y\) is a unit, and (U) means invertibility; (2) strongly stable: \(y\) is a unit times an element which is idempotent modulo the Jacobson radical, and (U) means invertibility; (3) weakly stable: (U) means one-sided invertibility; (4) generalized stable: (U) means there are \(s,t\in R\) such that \(s(a+by)t=1\); (5) QB-ring: (U) means \(a+by=u\) satisfies \((1-uc)R(1-du)=(1-du)R(1-uc)=0\) for some \(c,d\in R\); (6) PB-ring: (U) means that for \(a+by=u\), the ideal \(R(1-uc)R(1-du)R\) is nilpotent for some \(c,d\in R\); (7) power-substitution: (U) means \(y\) is an \(n\times n\) matrix over \(R\) and \(aI_n+by\) is invertible. Equivalent versions of these conditions are established, classes of rings where they hold are given, and applications such as appropriate forms of cancellation and diagonal reduction are proved.

Other chapters of the book deal with higher stable range conditions; versions of stable range \(1\) for two-sided ideals; cleanness (every element of the ring is a sum of an idempotent and a unit); \(K_0\) of exchange rings whose idempotents are central. In addition to cancellation and diagonal reduction, recurring applications to comparability of modules, structure of algebraic \(K_1\), and matrix factorization are developed.

The fundamental results in which cancellation is obtained entirely from stable range conditions were established by E. G. Evans, jun. [Pac. J. Math. 46, 115-121 (1973; Zbl 0272.13006)] and R. B. Warfield, jun. [Pac. J. Math. 91, 457-485 (1980; Zbl 0484.16017)]. Evans’ result states that if the endomorphism ring of a module \(A\) has stable range \(1\), then \(A\) cancels from direct sums: \(A\oplus B\cong A\oplus C\) implies \(B\cong C\). Warfield proved that if the endomorphism ring of \(A\) has stable range \(\leq n\) and \(B\) has a direct summand isomorphic to \(\bigoplus^nA\), then \(A\oplus B\cong A\oplus C\) implies \(B\cong C\). In the case of a von Neumann regular ring \(R\), L. Fuchs [Monatsh. Math. 75, 198-204 (1971; Zbl 0225.16013)] and I. Kaplansky [“Bass’s first stable range condition”, Mimeographed notes (1971)] (results incorporated in [L. N. Vaserstein, J. Pure Appl. Algebra 34, 319-330 (1984; Zbl 0547.16017)]) proved that \(R\) has stable range \(1\) if and only if it has internal cancellation: \(R=A_1\oplus B_1=A_2\oplus B_2\) (as modules) with \(A_1\cong A_2\) implies \(B_1\cong B_2\); this was extended to cancellation for finitely generated projective \(R\)-modules by D. Handelman [J. Algebra 48, 1-16 (1977; Zbl 0363.16009)]. Later, H.-P. Yu proved that an exchange ring \(R\) has stable range \(1\) if and only if the finitely generated projective \(R\)-modules have cancellation [J. Pure Appl. Algebra 98, No. 1, 105-109 (1995; Zbl 0837.16009)].

Diagonal reduction of matrices \(X\) (the existence of invertible matrices \(U\) and \(V\) such that \(UXV\) is diagonal) was first related to cancellation conditions in the case of a von Neumann regular ring \(R\), by P. Menal and J. Moncasi [J. Pure Appl. Algebra 24, 25-40 (1982; Zbl 0484.16006)], who proved that all (rectangular) matrices over \(R\) admit diagonal reduction if and only if \(R^2\oplus B\cong R\oplus C\) implies \(R\oplus B\cong C\) for all \(R\)-modules \(B\), \(C\). Ara, O’Meara, Pardo and the reviewer proved that over an exchange ring \(R\), all von Neumann regular square matrices admit diagonal reduction if and only if the finitely generated projective \(R\)-modules satisfy separative cancellation: \(A\oplus A\cong A\oplus B\cong B\oplus B\) implies \(A\cong B\) [P. Ara, K. R. Goodearl, K. C. O’Meara and E. Pardo, Linear Algebra Appl. 265, 147-163 (1997; Zbl 0883.15006)].

The classical stable range \(1\) condition for a unital ring \(R\) can be stated as follows: If \(a,b\in R\) and \(aR+bR=R\), there exists \(y\in R\) such that \(a+by\) is a unit. (It is known that this condition is left-right symmetric.) Many variations are investigated in the book, where “\(a+by\) is a unit” is replaced by “\(a+by\) has property (U)” and sometimes a restriction is imposed on \(y\). Here are some samples: (1) unit \(1\)-stable range: \(y\) is a unit, and (U) means invertibility; (2) strongly stable: \(y\) is a unit times an element which is idempotent modulo the Jacobson radical, and (U) means invertibility; (3) weakly stable: (U) means one-sided invertibility; (4) generalized stable: (U) means there are \(s,t\in R\) such that \(s(a+by)t=1\); (5) QB-ring: (U) means \(a+by=u\) satisfies \((1-uc)R(1-du)=(1-du)R(1-uc)=0\) for some \(c,d\in R\); (6) PB-ring: (U) means that for \(a+by=u\), the ideal \(R(1-uc)R(1-du)R\) is nilpotent for some \(c,d\in R\); (7) power-substitution: (U) means \(y\) is an \(n\times n\) matrix over \(R\) and \(aI_n+by\) is invertible. Equivalent versions of these conditions are established, classes of rings where they hold are given, and applications such as appropriate forms of cancellation and diagonal reduction are proved.

Other chapters of the book deal with higher stable range conditions; versions of stable range \(1\) for two-sided ideals; cleanness (every element of the ring is a sum of an idempotent and a unit); \(K_0\) of exchange rings whose idempotents are central. In addition to cancellation and diagonal reduction, recurring applications to comparability of modules, structure of algebraic \(K_1\), and matrix factorization are developed.

Reviewer: Kenneth R. Goodearl (Santa Barbara)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16E20 | Grothendieck groups, \(K\)-theory, etc. |

16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |

16S50 | Endomorphism rings; matrix rings |

16U60 | Units, groups of units (associative rings and algebras) |

16D80 | Other classes of modules and ideals in associative algebras |

19B10 | Stable range conditions |

15A21 | Canonical forms, reductions, classification |