Matrix semigroups

Author:
Mohan S. Putcha

Journal:
Proc. Amer. Math. Soc. **88** (1983), 386-390

MSC:
Primary 20M10

DOI:
https://doi.org/10.1090/S0002-9939-1983-0699399-0

MathSciNet review:
699399

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Abstract: Let be a semigroup of matrices over a field such that a power of each element lies in a subgroup (i.e., each element has a Drazin inverse within the semigroup). The main theorem of this paper is that there exist ideals of such that , is completely simple, and each Rees factor semigroup , , is either completely 0-simple or else a nilpotent semigroup. The basic technique is to study the Zariski closure of , which is a linear algebraic semigroup.

**[1]**G. Azumaya,*Strongly**-regular rings*, J. Fac. Sci. Hokkaido Univ.**13**(1954), 34-39. MR**0067864 (16:788e)****[2]**W. E. Clark,*Remarks on the kernel of a matrix semigroup*, Czechoslovak Math. J.**15(90)**(1965), 305-310. MR**0177047 (31:1311)****[3]**A. H. Clifford and G. B. Preston,*The algebraic theory of semigroups*, Vol. 1, Math. Surveys, no. 7, Amer. Math. Soc., Providence, R.I., 1961. MR**0132791 (24:A2627)****[4]**M. P. Drazin,*Pseudo-inverses in associative rings and semigroups*, Amer. Math. Monthly**7**(1958), 506-514. MR**0098762 (20:5217)****[5]**C. Faith,*Algebra*. II.*Ring theory*, Springer-Verlag, Berlin and New York, 1976. MR**0427349 (55:383)****[6]**W. D. Munn,*Pseudo-inverses in semigroups*, Proc. Cambridge Philos. Soc.**57**(1961), 247-250. MR**0121410 (22:12148)****[7]**M. S. Putcha,*On linear algebraic semigroups*, Trans. Amer. Math. Soc.**259**(1980), 457-469. MR**567091 (81i:20087)****[8]**-,*On linear algebraic semigroups*. II, Trans. Amer. Math. Soc.**259**(1980), 471-491.**[9]**I. R. Shafarevich,*Basic algebraic geometry*, Springer-Verlag, Berlin and New York, 1977. MR**0447223 (56:5538)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0699399-0

Keywords:
Matrix semigroup,
algebraic semigroup,
nil,
nilpotent,
chain conditions

Article copyright:
© Copyright 1983
American Mathematical Society