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Hereditary and semihereditary rings without nilpotent elements. (English. Russian original) Zbl 0763.16006

Algebra Logic 29, No. 3, 192-203 (1990); translation from Algebra Logika 29, No. 3, 284-302 (1990).
The word “ring” will mean “associative ring”, not necessarily with unity. Recall that a ring with unity is called a right p.p. ring if each principal right ideal is projective. A ring is called semihereditary (hereditary) if each finitely generated right ideal (each right ideal) is projective. In this paper we will define the corresponding concepts for rings without unity (in the case of rings with unity these definitions agree with the original ones).

MSC:

16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
16D25 Ideals in associative algebras
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[1] R. Gonchigdorzh, ”The ring of regular quotients of a reduced ring,” Algebra Logika,26, No. 2, 150–164 (1987). · Zbl 0661.16002
[2] P. T. Johnstone, Topos Theory, Academic Press, New York (1977).
[3] R. Hartshorne, Algebraic Geometry [Russian translation], Mir, Moscow (1981). · Zbl 0532.14001
[4] F. Albrecht, ”On projective modules over semihereditary rings,” Proc. Am. Math. Soc.,12, 638–639 (1961). · Zbl 0118.04401 · doi:10.1090/S0002-9939-1961-0126470-X
[5] R. F. Arens and I. Kaplansky, ”Topological representation of algebras,” Trans. Am. Math. Soc.,63, 457–481 (1948). · Zbl 0032.00702 · doi:10.1090/S0002-9947-1948-0025453-6
[6] G. M. Bergman, ”Hereditary commutative rings and the center of hereditary rings,” Proc. London Math. Soc.,23, 214–236 (1971). · Zbl 0219.13018 · doi:10.1112/plms/s3-23.2.214
[7] W. D. Burgess and W. Stephenson, ”Pierce sheaves of noncommutative rings,” Commun. Algebra,4, 51–75 (1976). · Zbl 0318.16005 · doi:10.1080/00927877608822094
[8] W. D. Burgess and W. Stephenson, ”An analog of the Pierce sheaf for noncommutative rings,” Commun. Algebra,6, 863–886 (1978). · Zbl 0374.16017 · doi:10.1080/00927877808822272
[9] J. Dauna and K. H. Hofmann, ”The representation of biregular rings by sheaves,” Math. Z.,91, 103–123 (1966). · Zbl 0178.37003 · doi:10.1007/BF01110158
[10] S. Endo, ”Note on p.p. rings,” Nagoya Math. J.,17, 167–170 (1960). · Zbl 0117.02203
[11] M. Ohori, ”Some studies of generalized p.p. rings and hereditary rings,” Math. J. Okayama Univ.,27, 53–70 (1985). · Zbl 0594.16016
[12] R. S. Pierce, ”Modules over commutative regular rings,” Memoirs Am. Math. Soc., No. 70 (1967). · Zbl 0152.02601
[13] B. Stenström, Rings of Quotients, Springer, Berlin (1975).
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