Strict radical classes of associative rings

Stewart, Patrick N.

doi:10.1090/S0002-9939-1973-0313296-5

Strict radical classes of associative rings
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by Patrick N. Stewart

Proc. Amer. Math. Soc. 39 (1973), 273-278

DOI: https://doi.org/10.1090/S0002-9939-1973-0313296-5

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Abstract:

A class of rings is strongly hereditary if it is closed under taking subrings. Strict radical classes (that is, radical classes for which the class of semisimple rings is strongly hereditary) are studied, strongly hereditary strict radical classes are classified, and it is shown that the category of associative rings has no proper localizing subcategories in the sense of Šul’geĭfer.

References

V. A. Andrunakievič, Radicals of associative rings. II. Examples of special radicals, Mat. Sb. (N.S.) 55 (97) (1961), 329–346 (Russian). MR 0137737
V. A. Andrunakievič and Ju. M. Rjabuhin, Rings without nilpotent elements, and completely prime ideals, Dokl. Akad. Nauk SSSR 180 (1968), 9–11 (Russian). MR 0230760
Nathan Divinsky, Rings and radicals, Mathematical Expositions, No. 14, University of Toronto Press, Toronto, Ont., 1965. MR 0197489
B. J. Gardner, Radicals of abelian groups and associative rings, Acta Math. Acad. Sci. Hungar. 24 (1973), 259–268. MR 323817, DOI 10.1007/BF01958035
A. G. Kuroš, Radicals in group theory, Sibirsk. Mat. Ž. 3 (1962), 912–931 (Russian). MR 0143813
A. H. Livšic, Category-theoretic foundations of the duality of radicality and semi-simplicity, Sibirsk. Mat. Ž. 5 (1964), 319–336 (Russian). MR 0164997
E. G. Šul′geĭfer, Functorial characterization of strict radicals in categories, Sibirsk. Mat. Ž. 7 (1966), 1412–1421 (Russian). MR 0204494
E. G. Šul′geĭfer, Localizations and strongly hereditary strict radicals in categories. , Trudy Moskov. Mat. Obšč. 19 (1968), 271–301 (Russian). MR 0237600
Patrick N. Stewart, Semi-simple radical classes, Pacific J. Math. 32 (1970), 249–254. MR 255593
Gabriel Thierrin, Sur les idéaux complètement premiers d’un anneau quelconque, Acad. Roy. Belg. Bull. Cl. Sci. (5) 43 (1957), 124–132 (French). MR 87639