Right orders in full linear rings

O’Meara, K. C.

doi:10.1090/S0002-9947-1975-0360663-6

Right orders in full linear rings
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by K. C. O’Meara PDF

Trans. Amer. Math. Soc. 203 (1975), 299-318 Request permission

Abstract:

In this paper a right order $R$ in an infinite dimensional full linear ring is characterized as a ring satisfying: (1) $R$ is meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals of $R$ are right annihilator ideals, and each such right ideal is essentially finitely generated; (3) $R$ possesses a reducing pair (i.e. a pair $({\beta _1},{\beta _2})$ of elements for which ${\beta _1}R,{\beta _2}R$ and $\beta _1^r + \beta _2^r$ are large right ideals of $R$); (4) for each $a \in R$ with ${a^l} = 0,aR$ contains a regular element of $R$. A second characterization of $R$ is also given. This is in terms of the right annihilator ideals of $R$ which have the same (uniform) dimension as ${R_R}$. The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.

References

Vasily C. Cateforis, Flat regular quotient rings, Trans. Amer. Math. Soc. 138 (1969), 241–249. MR 238899, DOI 10.1090/S0002-9947-1969-0238899-1
Vasily C. Cateforis, On regular self-injective rings, Pacific J. Math. 30 (1969), 39–45. MR 248178, DOI 10.2140/pjm.1969.30.39
Carl Faith, Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR 0227206, DOI 10.1007/BFb0074319
George D. Findlay, Flat epimorphic extensions of rings, Math. Z. 118 (1970), 281–288. MR 280550, DOI 10.1007/BF01109864
A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220. MR 111766, DOI 10.1112/plms/s3-10.1.201
John J. Hutchinson, Quotient full linear rings, Proc. Amer. Math. Soc. 28 (1971), 375–378. MR 424867, DOI 10.1090/S0002-9939-1971-0424867-8
R. E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891–895. MR 45695, DOI 10.1090/S0002-9939-1951-0045695-9
R. E. Johnson, Structure theory of faithful rings. III. Irreducible rings, Proc. Amer. Math. Soc. 11 (1960), 710–717. MR 118739, DOI 10.1090/S0002-9939-1960-0118739-9
R. E. Johnson, Quotient rings of rings with zero singular ideal, Pacific J. Math. 11 (1961), 1385–1392. MR 143779, DOI 10.2140/pjm.1961.11.1385
R. E. Johnson, Rings of finite rank, Publ. Math. Debrecen 11 (1964), 284–287. MR 179205
Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
Y ichi Miyashita, On quasi-injective modules. A generalization of the theory of completely reducible modules, J. Fac. Sci. Hokkaido Univ. Ser. I 18 (1964/1965), 158–187. MR 0171817
Kiiti Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971), 25–40. MR 286833, DOI 10.1007/BF01109715
K. C. O’Meara, Primeness of right orders in full linear rings, J. Algebra 26 (1973), 172–184. MR 332857, DOI 10.1016/0021-8693(73)90018-5
K. C. O’Meara, Intrinsic extensions of prime rings, Pacific J. Math. 51 (1974), 257–269. MR 480608, DOI 10.2140/pjm.1974.51.257
Nicolae Popescu and Tiberiu Spircu, Quelques observations sur les épimorphismes plats (à gauche) d’anneaux, J. Algebra 16 (1970), 40–59 (French). MR 265413, DOI 10.1016/0021-8693(70)90039-6
Nicolae Popescu and Dorian Spulber, Sur les quasi-ordres (à gauche) dans un anneau, J. Algebra 17 (1971), 474–481 (French). MR 274519, DOI 10.1016/0021-8693(71)90004-4
L. Silver, Noncommutative localizations and applications, J. Algebra 7 (1967), 44–76. MR 217114, DOI 10.1016/0021-8693(67)90067-1
Y. Utumi, On rings of which any one-sided quotient rings are two-sided, Proc. Amer. Math. Soc. 14 (1963), 141–147. MR 142568, DOI 10.1090/S0002-9939-1963-0142568-6