monography by Bohdan Zabavsky

Contents

1. Bass stable range

1.1. Stable range conditions
1.2. Ring theoretical constructions and their stable range
1.3. Completion to invertible matrix
1.4. Stabilization in K-theory
1.5. Examples and stable range calculations

2. Rings ruled by units, annihilators and idempotents

2.1. Von Neumann regular rings
2.2. Exchange, clean and idempotent stable range one rings
2.3. Potent and semipotent rings
2.4. VNL, NJ and semiregular rings
2.5. PM and Gelfand rings
2.6. Semihereditary and morphic rings

3. Rings defined by matrix canonical forms

3.1. Hermite and elementary divisor rings
3.2. Bezout rings and Shores test
3.3. Stable range and matrix diagonalization
3.4. Pullbacks and D+M-construction

4. Adequacy

4.1. Adequate and coadequate elements
4.2. Stable range of adequate rings
4.3. Zero-adequate and everywhere adequate rings

5. Finite homomorphic images

5.1. Minimal prime spectrum and fractionally regular rings
5.2. Semiregularity of zero-adequate rings
5.3. Avoidable rings
5.4. Effective and Dirichlet rings
5.5. Neat range one
5.6. Meaningful ring
5.7. Bezout morphic rings and units lifting
5.8. Gelfand range one and Bezout PM*-domains
5.9. Lattice-ordered groups and Montgomery counterexample
5.10. Rings of continuous functions C(X)

6. Bezout domains and their overrings

6.1. Almost stable range one
6.2. Finite localizing embeddings
6.3. Full matrices over elementary divisor rings
6.4. Sharp Bezout domains

7. Diagonalization over noncommutative rings

7.1. Simple Ore and Bezout rings
7.2. Idempotent matrix diagonalization
7.3. Distributive Bezout rings and Dubrovin condition
7.4. When GLn(R) is closed under transposition
7.5. Right Bezout rings and unimodular rows

8. Rings defined by range conditions

8.1. Neat range one and stable range
8.2. Semihereditary and von Neumann regular range one
8.3. Additive range one
8.4. Dyadic range one

9. Problems related to range conditions

9.1. Rings ruled by annihilators and idempotents
9.2. Commutative Bezout Rings
9.3. Nontrivial Finite Homomorphic Images of Commutative Bezout Rings
9.4. Adequate Rings and Their Generalizations
9.5. Local properties and ranges of commutative rings
9.6. Noncommutative Bezout rings
9.7. Ultimate problems