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... However, Rx 5 R, and so x is not unimodular. Chakravarti showed in [2] how the previous definition should be modified to handle FBN rings. To deal with any right and left noetherian ring, a different definition is needed. ...

In this paper we propose a notion of basic element for noncommutative noetherian rings and derive the corresponding Basic Element Theorem as in (D. Eisenbud and E. G. Evans Jr., Generating modules efficiently: Theorems from algebraic K-theory, J. Algebra27 (1973), 278–305). As corollaries to this theorem we obtain a number of results of Stafford, such as his generalizations of Serre's Splitting-Off Theorem and Bass's Cancellation Theorem.

The well-known Forster-Swan theorem, and the stable version of this result given by Eisenbud and Evans, give a bound on the number of generators of a module over a commutative ring in terms of local data. In this paper we generalise these results to arbitrary Noetherian rings. As an application we prove versions of Serre's Theorem and the Cancellation Theorem over such rings.

Rings of Quotients: An Introduction to Methods of Ring Theory The Number of Generators of a Module over a Fully Bounded Ring

- B Stenstrom
- R B Warfield
- Jr

B. Stenstrom, "Rings of Quotients: An Introduction to Methods of Ring Theory," Springer-Verlag, BerlinIHeidelberglNew York, 1975. R. B. Warfield, Jr., The Number of Generators of a Module over a Fully Bounded Ring, J. Algebra 66 (1980), 425-447.

Commutative Algebra Addison-Wesley, Lon-don, 1972. M. Chamarie, Modules sur les anneaux de Krull non commu-tatifs in "Seminaire d'Algebre P Generating Modules Efficiently: Theorems from Algebraic K -Theory

- N Bourbaki
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- E G Evans

N. Bourbaki, "Commutative Algebra," Addison-Wesley, Lon-don, 1972. M. Chamarie, Modules sur les anneaux de Krull non commu-tatifs in "Seminaire d'Algebre P. Dubreil et M.-P. Malliavin," Lecture Notes in Math. 1029, Springer-Verlag, BerlinIHeidelberdNew York, 1983. D. Eisenbud and E.G. Evans, Generating Modules Efficiently: Theorems from Algebraic K -Theory, J. Algebra 27 (1973), 278-305.

Modules Projectifs et Espaces Fibres a Fibre Vec-torielle Cancellation for non-projective modules in "Module Theory

- G Maury
- J Raynaud Maximaux Au Sens
- K Asano
- J.-P Serre

G. Maury and J. Raynaud, "Ordres Maximaux au sens de K. Asano," Lecture Notes in Math. 808, Springer-Verlag, Berlin/Heidelberg/New York, 1980. J.-P. Serre, "Modules Projectifs et Espaces Fibres a Fibre Vec-torielle," Seminaire P. Dubreil, Expose 23 (1957-58), Paris. J. T. Stafford, Cancellation for non-projective modules in "Module Theory, Proceedings, Seattle, 1977," Lecture Notes Downloaded by [Michigan State University] at 00:03 14 February 2015 in Math. 700, Springer-Verlag, Berlin/Heidelberg/New York, 1979. J. T. Stafford, Generating Modules Efficiently: Algebraic K -Theory for Noncommutative Noetherian Rings, J. Algebra 69 (1981), 312-346.