Later developments based on some ideas of Andrunachievici: Special radicals and The Lemma
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2018), pp. 113-137.

Voir la notice de l'article provenant de la source Math-Net.Ru

This is a survey of subsequent work on two topics derived from fundamental publications of V. A. Andrunachievici in the 1950s and 1960s: special radicals and the result which has come to be known as the Andrunachievici Lemma.
@article{BASM_2018_2_a9,
     author = {B. J. Gardner},
     title = {Later developments based on some ideas of {Andrunachievici:} {Special} radicals and {The} {Lemma}},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {113--137},
     publisher = {mathdoc},
     number = {2},
     year = {2018},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/BASM_2018_2_a9/}
}
TY  - JOUR
AU  - B. J. Gardner
TI  - Later developments based on some ideas of Andrunachievici: Special radicals and The Lemma
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2018
SP  - 113
EP  - 137
IS  - 2
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/BASM_2018_2_a9/
LA  - en
ID  - BASM_2018_2_a9
ER  - 
%0 Journal Article
%A B. J. Gardner
%T Later developments based on some ideas of Andrunachievici: Special radicals and The Lemma
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2018
%P 113-137
%N 2
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/BASM_2018_2_a9/
%G en
%F BASM_2018_2_a9
B. J. Gardner. Later developments based on some ideas of Andrunachievici: Special radicals and The Lemma. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2018), pp. 113-137. https://geodesic-test.mathdoc.fr/item/BASM_2018_2_a9/

[1] Andrunakievič V. A., “Radicals of associative rings, I”, Amer. Math. Soc. Transl. (2), 52 (1966), 95–128 ; Mat. Sb., 44(86):2 (1958), 179–212 ; “II”, Amer. Math. Soc. Transl. (2), 52 (1966), 129–149 ; Mat. Sb., 55(97):3 (1961), 329–346 | Zbl | MR | Zbl | Zbl | MR | Zbl

[2] Andrunakievich V. A., Ryabukhin Yu. M., “Torsions and Kurosh chains in algebras”, Trudy Moskov. Mat. Obshchestva, 29, 1973, 19–49 (in Russian) | MR | Zbl

[3] Gardner B. J., Wiegandt R., Radical theory of rings, Marcel Dekker, New York–Basel, 2004 | MR | Zbl

[4] Gardner B. J., Radical theory, Longman, Harlow, 1989 | MR | Zbl

[5] Heyman G. A. P., Roos C., “Essential extensions in radical theory for rings”, J. Aust. Math. Soc., 23 (1977), 340–347 | DOI | MR | Zbl

[6] McKnight J. D., Musser G. L., “Special $(p,q)$ radicals”, Canad. J. Math., 24 (1972), 38–44 | DOI | MR | Zbl

[7] Beidar K. I., “The intersection property for radicals”, Usp. Mat. Nauk., 44:1(265) (1989), 187–188 (in Russian) | MR | Zbl

[8] Ryabukhin Yu. M., “On supernilpotent and special radicals”, Issled. Alg. Mat. Anal., Kishinev, 1965, 65–72 | MR

[9] Ryabukhin Yu. M., “Supernilpotent and special radicals”, Mat. Issled., 48 (1978), 80–93 (in Russian) | MR | Zbl

[10] Rossa R. F., “More properties inherited by the lower radical”, Proc. Amer. Math. Soc., 33 (1972), 247–249 | DOI | MR | Zbl

[11] Beidar K. I., Salavová K., “Some examples of supernilpotent nonspecial radicals”, Acta Math. Hungar., 40 (1982), 109–112 | DOI | MR | Zbl

[12] Ryabukhin Yu. M., “Incomparable nilradicals and nonspecial hypernilpotent radicals”, Algebra and Logic, 14 (1975), 54–63 | DOI | MR

[13] Rowen L. H., “A subdirect decomposition of semiprime rings and its application to maximal quotient rings”, Proc. Amer. Math. Soc., 46 (1974), 176–188 | DOI | MR

[14] Gardner B. J., Stewart P. N., “Prime essential rings”, Proc. Edinburgh Math. Soc. (2), 34 (1991), 241–250 | DOI | MR | Zbl

[15] France-Jackson H., “On prime essential rings”, Bull. Aust. Math. Soc., 47 (1993), 287–290 | DOI | MR | Zbl

[16] France-Jackson H., “On bad supernilpotent radicals”, Bull. Aust. Math. Soc., 85 (2012), 271–274 | DOI | MR | Zbl

[17] Jaegermann M., Sands A. D., “On normal radicals, $N$-radicals, and $A$-radicals”, J. Algebra, 50 (1978), 337–349 | DOI | MR | Zbl

[18] Stewart P. N., “Strongly hereditary radical classes”, J. London Math. Soc. (2), 4 (1972), 499–509 | DOI | MR | Zbl

[19] Stewart P. N., “Radicals and functional representations”, Acta. Math. Acad. Sci. Hungar., 27 (1976), 319–321 | DOI | MR | Zbl

[20] Gardner B. J., “Some recent results and open problems concerning special radicals”, Radical Theory, Proc. (1988, Sendai Conference), Uchida Rokakuho, Tokyo, 1988, 25–56 | MR

[21] Snider R. L., “Lattices of radicals”, Pacific J. Math., 40 (1972), 207–220 | DOI | MR | Zbl

[22] Gardner B. J., Wiegandt R., “Characterizing and constructing special radicals”, Acta Math. Acad. Sci. Hungar., 40 (1982), 73–83 | DOI | MR | Zbl

[23] Krachilov K. K., “Coatoms in the lattice of special radicals”, Mat. Issled., 49 (1979), 80–86 (in Russian) | MR | Zbl

[24] Andrunakievich V. A., Ryabukhin Yu. M., Radicals of algebras and structure theory, Nauka, Moscow, 1979 (in Russian) | MR

[25] Korolchuk H., “A note on the lattice of special radicals”, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 29 (1981), 103–104 | MR

[26] Liang Zhian, “A note on the atoms of the lattice of special radical classes”, Acta Sci. Natur. Univ. NeiMenggu, 24 (1993), 480–483 | MR | Zbl

[27] France-Jackson H., “On special atoms”, J. Aust. Math. Soc. Ser. A, 64 (1998), 302–306 | DOI | MR | Zbl

[28] France-Jackson H., Groenewald N. J., “On rings generating supernilpotent and special atoms”, Quaest. Math., 28 (2005), 471–478 | DOI | MR | Zbl

[29] Leavitt W. G., van Leeuwen L. C. A., “Rings isomorphic with all proper factor-rings”, Ring Theory, Proc. Antwerp Conference (Univ. Antwerp., 1978), Lecture Notes in Pure and Appl. Math., 51, Marcel Dekker, New York, 1978, 783–798 | MR

[30] Wahyuni S., Wijayanti I. E., France-Jackson H., “Prime essential ring that generates a special atom”, Bull. Aust. Math. Soc., 95 (2017), 214–218 | DOI | MR | Zbl

[31] Gardner B. J., “Prime rings for which the set of nonzero ideals is a special class”, J. Aust. Math. Soc. Ser. A, 51 (1991), 27–32 | DOI | MR | Zbl

[32] Andruszkiewicz R. R., “The classification of integral domains in which the relation of being an ideal is transitive”, Comm. Algebra, 31 (2003), 2067–2093 | DOI | MR | Zbl

[33] Gardner B. J., “Injectives for ring monomorphisms with accessible images”, Comm. Algebra, 10 (1982), 673–694 | DOI | MR | Zbl

[34] Gardner B. J., Stewart P. N., “Injectives for ring monomorphisms with accessible images, II”, Comm. Algebra, 13 (1985), 133–145 | DOI | MR | Zbl

[35] Feigelstock S., Additive groups of rings, Pitman, Boston–London–Melbourne, 1983 | MR | Zbl

[36] Puczyłowski E. R., Roszkowska E., “Atoms of lattices of radicals of associative rings”, Radical Theory, Proc. (1988, Sendai Conference), Uchida Rokakuho, Tokyo, 1988, 123–134 | MR

[37] France-Jackson H., “On rings generating atoms of lattices of special and supernilpotent radicals”, Bull. Aust. Math. Soc., 44 (1991), 203–205 | DOI | MR | Zbl

[38] Gardner B. J., “Polynomial identities and radicals”, Compositio Math., 35 (1977), 269–279 | MR | Zbl

[39] Osborn J. M., “Varieties of algebras”, Adv. Math., 8 (1972), 163–369 | DOI | MR | Zbl

[40] Leavitt E. G., “A minimally embeddable ring”, Period. Math. Hungar., 12 (1981), 129–140 | DOI | MR | Zbl

[41] Stewart P. N., “Semi-simple radical classes”, Pacific J. Math., 32 (1970), 249–254 | DOI | MR | Zbl

[42] Beidar K. I., “On questions of B. J. Gardner and A. D. Sands”, J. Aust. Math. Soc. Ser. A, 56 (1994), 314–319 | DOI | MR | Zbl

[43] France-Jackson H., “$\ast$-rings and their radicals”, Quaest. Msath., 8 (1985), 231–239 | DOI | MR | Zbl

[44] Vodyanyuk E. A., “Uniquely defined special radicals”, Mat. Issled., 74 (1983), 18–29 | MR | Zbl

[45] France-Jackson H., “On similar special classes”, Acta. Math. Hungar., 93 (2001), 249–251 | DOI | MR | Zbl

[46] Shatalova M. A., “$\ell_A$- and $\ell_I$-rings”, Siberian Math. J., 4 (1966), 1084–1095 | DOI | MR

[47] Shatalova M. A., “The theory of radicals in lattice-ordered rings”, Math. Notes, 4 (1968), 875–880 | DOI | Zbl

[48] Steinberg S. A., “Radical theory in lattice-ordered rings”, Symposia Mathematica XXI, Convegno sulle Misure su Gruppi e su Spazi Vectoriali, Convegno sulli Gruppi e Anelli Ordinati (INDAM, Rome, 1975), Academic Press, London, 1977, 379–400 | MR

[49] Shavgulidze N. E., “Special classes of $\ell$-rings”, J. Math. Sci. (N.Y.), 166:6 (2010), 794–805 ; Fundam. Prikl. Mat., 15:1 (2009), 157–173 | DOI | MR | Zbl

[50] Shavgulidze N. E., “Special classes of $\ell$-rings and the Anderson–Divinsky–Suliński lemma”, Moscow Univ. Math. Bull., 65:2 (2010), 76–77 | DOI | MR | Zbl

[51] Salavová K., “Radicals of rings with involution. I”, Comment. Math. Univ. Carolinae, 18 (1977), 367–381 (in Russian) | MR | Zbl

[52] Salavová K., “Radicals of rings with involution. II”, Comment. Math. Univ. Carolinae, 18 (1977), 455–466 (in Russian) | MR | Zbl

[53] Booth G. L., Groenewald N. J., “Special radicals in rings with involution”, Publ. Math. Debrecen, 48 (1996), 241–251 | MR | Zbl

[54] Booth G. L., “On lattices of radicals of involution rings”, Sci. Math. Jpn., 63 (2006), 387–394 | MR | Zbl

[55] Balaba I. N., “Special radicals of graded rings”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2004, no. 1, 26–33 | MR | Zbl

[56] Ilić-Georgijević E., Vuković M., “A note on general radicals of paragraded rings”, Sarajevo J. Math., 12(25):2, suppl. (2016), 317–324 | MR

[57] Krasner M., “Anneaux gradués généraux”, Publications des Séminaires de Mathḿatiques et Informatique de Rennes, Fasc. S3 “Colloque d'Algèbre”, 1980, 209–308 | MR | Zbl

[58] Ryabukhin Yu. M., “Radicals in $\Omega$-groups. III. Special and quasi-special radicals”, Mat. Issled., 1(11) (1969), 110–131 (in Russian) | MR | Zbl

[59] Buys A., Gerber G. K., “Special classes in $\Omega$-groups”, Ann. Univ. Sci. Budapest. Sect. Math., 29 (1986), 73–85 | MR | Zbl

[60] Booth G. L., Groenewald N. J., “Special radicals of $\Omega$-groups”, Nearrings, nearfields and $K$-loops (Hamburg, 1995), Math. Appl., 426, Kluwer Acad. Publ., Dordrecht, 1997, 211–218 | MR | Zbl

[61] Kaarli K., “Special radicals of near-rings”, Tartu Riikl. Ul. Toimetised, 610 (1982), 53–68 (in Russian) | MR | Zbl

[62] Booth G. L., Groenewald N. J., “Special radicals of near-rings”, Math. Japon., 37 (1992), 701–706 | MR | Zbl

[63] Birkenmeier G. F., Heatherly H. E., Lee E. K. S., “Special radicals for near-ring”, Tamkang. J. Math., 27 (1996), 281–288 | MR | Zbl

[64] Groenewald N. J., “Different prime ideals in near-rings”, Comm. Algebra, 19 (1991), 2667–2675 | DOI | MR | Zbl

[65] Suliński A., “The Brown–McCoy radical in categories”, Fund. Math., 59 (1966), 23–41 | DOI | MR | Zbl

[66] Gardner B. J., “Transfer properties in radical theory”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2004, no. 1, 46–56 | MR | Zbl

[67] Andrunakievich V. A., Ryabukhin Yu. M., “The existence of the Brown–McCoy radical in Lie algebras”, Soviet Math. Dokl., 9 (1968), 373–376 | Zbl

[68] Pilz G. F., Near-rings. The theory and its applications, Second edition, North-Holland, Amsterdam, 1983 | MR | Zbl

[69] Kaarli K., “On Jacobson type radicals of near-rings”, Acta. Math. Hungar., 50 (1987), 71–78 | DOI | MR | Zbl

[70] Birkenmeier G., Heatherly H., “Minimal ideals in near-rings”, Comm. Algebra, 20 (1992), 457–468 | DOI | MR | Zbl

[71] Birkenmeier G., Heatherly H., “Minimal ideals in near-rings and localized distributivity conditions”, J. Aust. Math. Soc. Ser. A, 54 (1993), 156–168 | DOI | MR | Zbl

[72] Beidar K. I., Fong Y., Shum K. P., “On the hearts of subdirectly irreducible near-rings”, Southeast Asian Bull. Mathematics, 18:2 (1994), 5–9 | MR | Zbl

[73] Birkenmeier G. F., Heatherly H. E., Lee E. K. S., “An Andrunakievich lemma for near-rings”, Comm. Algebra, 23 (1995), 2825–2850 | DOI | MR | Zbl

[74] Hentzel I. R., Slater M., “On the Andrunakievich lemma for alternative rings”, J. Algebra, 27 (1973), 243–256 | DOI | MR | Zbl

[75] Pchelintsev S. V., “Meta-ideals of alternative algebras”, Siberian Math. J., 24 (1983), 433–439 | DOI | MR

[76] Hentzel I. R., “The Andrunakievich lemma for alternative rings”, Algebras Groups Geom., 6 (1989), 55–64 | MR | Zbl

[77] Zhevlakov K. A., Slin'ko A. M., Shestakov I. P., Shirshov A. I., Rings that are nearly associative, Academic Press, New York–London, 1982 ; Nauka, Moscow, 1978 | MR | Zbl | Zbl

[78] Slin'ko A. M., “On radical Jordan rings”, Algebra and Logic, 11 (1972), 121–126 | DOI

[79] Medvedev Yu. A., “An analogue of the Andrunakievich lemma for Jordan algebras”, Siberian Math J., 28:6 (1987), 928–936, Russian original: pp. | DOI | MR | Zbl

[80] Slater M., “On the Andrunakievich lemma for linear Jordan rings”, Algebra and Logic, 26 (1987), 69–78 | DOI | MR | Zbl

[81] Beidar K. I., “The Andrunakievich lemma and Jordan algebras”, Russian Math. Surveys, 45 (1990), 159 | DOI | MR | Zbl

[82] Nikitin A. A., “Heredity of radicals of rings”, Algebra and Logic, 17 (1978), 210–217 | DOI | MR

[83] Skosyrskii V. G., “Radicals of Jordan algebras”, Siberian Math. J., 29 (1988), 283–293 | DOI | MR | Zbl

[84] Anquela J. A., Cortés T., “Minimal ideals of Jordan systems”, Invent. Math., 168 (2007), 83–90 | DOI | MR | Zbl

[85] Anquela J. A., Cortés T., McCrimmon K., “Trivial minimal ideals of Jordan systems”, J. Algebra, 328 (2011), 167–177 | DOI | MR | Zbl

[86] Anderson T., Gardner B. J., “Semi-simple classes in a variety satisfying an Andrunakievich lemma”, Bull. Aust. Math. Soc., 18 (1978), 187–200 | DOI | MR | Zbl

[87] Ánh P. N., Loi N. V., Wiegandt R., “On the radical theory of Andrunakievich varieties”, Bull. Aust. Math. Soc., 31 (1985), 257–269 | DOI | MR | Zbl