On the structure of semiprime rings
Author: Augusto H. Ortiz
Journal: Proc. Amer. Math. Soc. 38 (1973), 22-26
MSC: Primary 16A12; Secondary 16A21
MathSciNet review: 0313292
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Abstract: The structure of prime rings has recently been studied by A. W. Goldie, R. E. Johnson, L. Lesieur and R. Croisot. In their main results some sort of finiteness assumption is invariably made. It is shown in this paper that certain semiprime rings are subdirect sums of full rings of linear transformations of a right vector space over a division ring. No finiteness assumption is made about the ring. An apparently new radical property is defined and some of its properties are established; e.g., the radical of a matrix ring is the matrix ring of the radical of .
-  V. A. Andrunakievič and Ju. M. Rjabuhin, Modules and radicals, Soviet Math. Dokl. 5 (1964), 728–731. MR 0190182
-  T. Chi-Te, Report on injective modules, Queen's Papers in Pure and Appl. Math., no. 6, Queen's University, Kingston, Ontario, 1966.
-  László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
-  Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR 0222106
-  Neal H. McCoy, The theory of rings, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1964. MR 0188241
-  Edward T. Wong, Endomorphisms of the quasi-injective hull of a module, Canad. Math. Bull. 13 (1970), 149–150. MR 0260790, https://doi.org/10.4153/CMB-1970-034-x
- V. A. Andrunakievič and Ju. M. Rjabuhin, Modules and radicals, Dokl. Akad. Nauk SSSR 156 (1964), 991-994 = Soviet Math. Dokl. 5 (1964), 728-731. MR 32 #7596. MR 0190182 (32:7596)
- T. Chi-Te, Report on injective modules, Queen's Papers in Pure and Appl. Math., no. 6, Queen's University, Kingston, Ontario, 1966.
- L. Fuchs, Infinite abelian groups. Vol. I, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970. MR 41 #333. MR 0255673 (41:333)
- N. Jacobson, Structure of rings, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1964. MR 36 #5158. MR 0222106 (36:5158)
- N. H. McCoy, The theory of rings, Macmillan, New York, 1964. MR 32 #5680. MR 0188241 (32:5680)
- E. T. Wong, Endomorphims of the quasi-injective hull of a module, Canad. Math. Bull. 13 (1970), 149-150. MR 0260790 (41:5413)
Keywords: Finiteness assumption, subdirect sums, radical property, primitive ideals, prime ideals, irreducible module, the prime radical, the Jacobson radical, hereditary radical, Prüfer module, essential submodule, small submodule, subdirectly irreducible, heart of a module, injective hull, quasi-injective hull, essential extension
Article copyright: © Copyright 1973 American Mathematical Society