On sums of diagonal and invertible formal matrices
This paper concerns properties of k-good formal matrix rings Kn of order n with rings R1, R2, ..., R
n on the main diagonal and R
rR,-bimodules My on other places. In the ring theory, various matrix rings play an important role. Above all I mean formal matrix rings. Formal matrix rings generalize a notion of matrix ring of order n over a given ring. Every ring with nontrivial idem-potents is isomorphic to some formal matrix ring. The endomorphism ring of a decomposable module also is a formal matrix ring. The studies of such rings are quite useful for solving some problems on endomorphism rings of Abelian groups. In this paper I show that every matrix form Kn is the sum of diagonal matrix and invertible matrix. Also I give one condition when Kn is the k-good ring.
Keywords
кольцо,
обобщенная матрица,
формальная матрица,
k-хорошее кольцо,
ring,
generalized matrix,
formal matrix,
k-good ringAuthors
| Norbosambuev Tsyrendorji D. | Tomsk State University | NsTsdDts@yandex.ru |
Всего: 1
References
Крылов П.А., Туганбаев А.А. Модули над кольцами формальных матриц // Фундаментальная и прикладная математика. 2009. Т. 15. № 8. С. 145-211.
Крылов П.А. О группе K0 кольца обобщeнных матриц // Алгебра и логика. 2013. Т. 52. № 3. C. 370-385.
Tang G., Zhou Y. A class of formal matrix rings // Linear Algebra and Appl. 2013. V. 428. P. 4672-4688.
Wolfson K.G. An ideal theoretic characterization of the ring of all linear transformations // Amer. J. Math. 1953. V. 75. P. 358-386.
Zelinsky D. Every linear transformation is a sum of non-singular ones // Proc. Amer. Math. Soc. 1954. V. 5. P. 627-630.
Skomyakov L. Complemented modular lattices and regular rings. London: Oliver&Boyd, 1958. 182 p.
Raphael R.M. Rings which are generated by their units // J. Algebra. 1974. V. 28. P. 199204.
Fuchs L. Recent results and problems on Abelian groups // Topics in Abelian groups (Proc. Sympos., New Mexico State University). 1962. Scott, Foresman, Chicago. P. 9-40.
Stringall R.W. Endomorphism rings of Abelian groups generated by automorphism groups // Acta Math. Acad. Sci. Hungar. 1967. V. 18. P. 401-404.
Freedman H. On endomorphisms of primary Abelian groups // J. London Math. Soc. 1968. V. 43. P. 305-307.
Hill P. Endomorphism ring generated by units // Trans. Amer. Math. Soc. 1969. V. 141. P. 99-105.
Castagna F. Sums of automorphisms of a primary Abelian group // Pacific J. Math. 1968. V. 27. P. 463-473.
Henriksen M. Two classes of rings generated by their units // J. Algebra. 1974. V. 31. P. 182193.
Vamos P. 2-good rings // Quart. J. Math. 2005. V. 56. P. 417-430.
Srivastava A.K. A survey of rings generated by units // Annales de la Faculte des Sciences de Toulouse Mathatiques. 2010. V. 19. P. 203-213.
