Geodesic
Endomorphism rings of free modules
Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 215-231 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Suppose $\mathfrak a$ is some property of modules. Let $\mathfrak{R_a}$ denote the class of rings over which all modules possess property $\mathfrak a$. The main theorem of this paper answers the following question for a rather extensive class of properties $\mathfrak a$; what must the property $\mathfrak b$ of modules be in order that $R\in\mathfrak{R_a}$ if and only if $\operatorname{End}_R(F)\in\mathfrak{R_b}$, for any free $R$-module $F$? Among the corollaries are many well-known theorems relating properties of the ring $R$ and the rings $\operatorname{End}_R(F)$, and also a number of new results of similar type. Bibliography: 35 titles. 
@article{SM_1974_23_2_a3,
     author = {G. M. Brodskii},
     title = {Endomorphism rings of free modules},
     journal = {Sbornik. Mathematics},
     pages = {215--231},
     year = {1974},
     volume = {23},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1974_23_2_a3/}
}
TY  - JOUR
AU  - G. M. Brodskii
TI  - Endomorphism rings of free modules
JO  - Sbornik. Mathematics
PY  - 1974
SP  - 215
EP  - 231
VL  - 23
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1974_23_2_a3/
LA  - en
ID  - SM_1974_23_2_a3
ER  - 
%0 Journal Article
%A G. M. Brodskii
%T Endomorphism rings of free modules
%J Sbornik. Mathematics
%D 1974
%P 215-231
%V 23
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1974_23_2_a3/
%G en
%F SM_1974_23_2_a3
G. M. Brodskii. Endomorphism rings of free modules. Sbornik. Mathematics, Tome 23 (1974) no. 2, pp. 215-231. http://geodesic.mathdoc.fr/item/SM_1974_23_2_a3/

[1] G. M. Brodskii, “Koltsa endomorfizmov svobodnykh modulei nad sovershennymi koltsami”, Matem. sb., 88 (130) (1972), 137–147 | MR | Zbl

[2] G. M. Brodskii, “O krucheniyakh v modulyakh”, Matem. zametki, 14:4 (1973), 527–534 | MR | Zbl

[3] I. Bukur, A. Delyanu, Vvedenie v teoriyu kategorii i funktorov, izd-vo «Mir», Moskva, 1972 | MR

[4] N. Burbaki, Algebra. Moduli, koltsa, formy, izd-vo «Nauka», Moskva, 1966 | MR

[5] N. Burbaki, Kommutativnaya algebra, izd-vo «Mir», Moskva, 1971 | MR

[6] V. I. Gemintern, “Samoin'ektivnye koltsa endomorfizmov svobodnykh modulei”, Matem. zametki, 6:5 (1969), 533–540 | MR

[7] V. E. Govorov, “Maloin'ektivnye moduli”, Algebra i logika, 2:6 (1963), 21–50 | MR | Zbl

[8] N. Dzhekobson, Stroenie kolets, IL, Moskva, 1961

[9] A. Kartan, S. Eilenberg, Gomologicheskaya algebra, IL, Moskva, 1960

[10] I. Lambek, Koltsa i moduli, izd-vo «Mir», Moskva, 1971 | MR

[11] S. Maklein, Gomologiya, izd-vo «Mir», Moskva, 1966

[12] A. P. Mishina, L. A. Skornyakov, Abelevy gruppy i moduli, izd-vo «Nauka», Moskva, 1969

[13] L. A. Skornyakov, “Gomologicheskaya klassifikatsiya kolets”, Matem. vestnik, 4:4 (1967), 415–434 | Zbl

[14] L. A. Skornyakov, Elementy teorii struktur, izd-vo «Nauka», Moskva, 1970 | MR

[15] V. Stefenson, G. M. Tsukerman, “Sib. matem. zh.”, Koltsa endomorfizmov proektivnykh modulei, 11:1 (1970), 228–232 | MR

[16] T. S. Tolskaya, “Kogda vse tsiklicheskie moduli suschestvenno vkladyvayutsya v svobodnye?”, Matem. issledovaniya, 5:2 (16) (1970), 187–192 | MR

[17] T. S. Tolskaya, “O koobrazuyuschikh koltsakh”, Matem. zametki, 10:6 (1971), 689–700

[18] G. M. Tsukerman, “Koltsa endomorfizmov svobodnykh modulei”, Sib. matem. zh., 7:5 (1966), 1161–1167 | Zbl

[19] G. M. Tsukerman, “O psevdoin'ektivnykh modulyakh i samopsevdoin'ektivnykh koltsakh”, Matem. zametki, 7:3 (1970), 369–380 | Zbl

[20] H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Trans. Amer. Math. Soc., 95:3 (1960), 466–488 | DOI | MR | Zbl

[21] S. Chase, “Direct products of modules”, Trans. Amer. Math. Soc., 97:3 (1960), 457–473 | DOI | MR

[22] P. M. Cohn, Morita-equivalence and duality, Math. Notes, Queen Mary College, London

[23] P. M. Cohn, “Some remarks on the invariant basis property”, Topology, 5:3 (1966), 215–228 | DOI | MR | Zbl

[24] C. Faith, “Rings with ascending condition on annihilators”, Nagoya Math. J., 27:1 (1966), 179–191 | MR | Zbl

[25] C. Faith, “On Köthe rings”, Math. Ann., 164:3 (1966), 207–212 | DOI | MR | Zbl

[26] C. Faith, E. Walker, “Direct sum representations of injective modules”, J. Algebra, 5:2 (1967), 203–221 | DOI | MR | Zbl

[27] P. Gabriel, N. Popescu, “Caractérisations des categories abéliennes avec générateurs et limites inductives exactes”, C. r. Acad. scient, 258 (1964), 4188–4190 | MR | Zbl

[28] M. Jkeda, T. Nakayama, “On some characteristic properties of quasi-Frobenius and regular rings”, Trans. Amer. Math. Soc., 5:1 (1954), 15–19 | MR

[29] R. E. Dohnson, E. T. Wong, “Quasi-injective modules and irreducible rings”, J. London Math. Soc., 36:3 (1961), 260–268 | DOI | MR

[30] T. Kato, “Duality of cyclic modules”, Tôhoku Math. J., 19:3 (1967), 349–356 | DOI | MR | Zbl

[31] W. G. Leavitt, “The module type of a ring”, Trans. Amer. Math. Soc., 103:1 (1962), 113–130 | DOI | MR | Zbl

[32] H. Lenzing, “Über kohärente Ringe”, Math. Z., 114:3 (1970), 201–212 | DOI | MR | Zbl

[33] H. Lenzing, “Halberbliche Endomorphismenringe”, Math. Z., 118:3 (1970), 219–240 | DOI | MR | Zbl

[34] F. L. Sandomierski, “Some examples of right self-injective rings which are not left self-injective”, Trans. Amer. Math. Soc., 26:2 (1970), 244–245 | MR | Zbl

[35] P. Vámos, “On ring classes defined by modules”, Publs. Math., 14:1–4 (1967), 1–8 | MR | Zbl