RECTANGULAR INVERTIBLE MATRICES
Keywords:
Rings with identity element, invertible matrices over a ring, invariant basis number, congruences.Abstract
Let n be a positive integer and R a ring with identity element, In this paper, invertibility condition of rectangular matrices over R are studied via congruence relations in N stipulating that rings have invariant basis number.
References
A. J. Berrick and M. E. Keating, Rectangular Invertible Matrices.
M. F. Atiyah and I. G. MacDonald, Introduction to Commutative
Algebra, Addison-Wesley, Reading, Mass. 1969.
G. M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc. 200 (1977), 33-88.
P. M. Cohn, Some remarks on the invariant basis property, Topology 5 (1966), 215-228.
P. M. Cohn, Algebra II, Wiley & Sons, Chichester, 1977.
P. M. Cohn, Universal Algebra, Reidel, Dordrecht, 1981.
P. M. Cohn, Algebra I, Wiley & Sons, Chichester, 1982.
K. Goodearl, P. Menal, and J. Moncasi, Free and residually artinian regular rings, J. Algebra 156 (1993), 407-432.
J. M. Howie, An Introduction to Semigroup Theory, London Math. Soc. Monograph 7, Academic Press, London, 1976.
W. van der Kallen, Injective stability for K2, Lecture Notes in Math. 551, Springer, Berlin, 1976, pp.77-154.
W. G. Leavitt, Modules without invariant basis number, Proc. Amer. Math. Soc. 8 (1957), 322-328.
W. G. Leavitt, The module type of a ring, Trans. Amer.
Math. Soc. 103 (1962), 113-130.
W. G. Leavitt, The module type of a homomorphic image, Duke Math. J. 32 (1965), 305-311.
L. N. Vaserstein , Stable ranks of rings and dimensionality
of topological spaces, Funct. Anal. And Appl. 5 (1971), 102-110.
Lous H. Rowen, Ring Theory, Volume I, 1.3 p. 61.