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Hereditary radicals and 0-bands of semigroups. (English) Zbl 0661.20043

With any abstract class \({\mathcal R}\) of semigroups is associated the map \(S\to \rho (S)\), the union of the ideals of S which belong to \({\mathcal R}\). This map is a “radical” (and \({\mathcal R}\) is a “radical class”) if (i) \({\mathcal R}\) is closed under Rees quotients, (ii) \(S\in {\mathcal R}\) if and only if \(\rho (S)=S\) and (iii) \(\rho (S/\rho (S))=\{0\}\) for any S. For example, the class \({\mathcal N}\) of nilsemigroups determines the “Clifford” radical and the class \({\mathcal L}\) of locally nilpotent semigroups determines the “Shevrin” radical.
The author studies the interaction between these, and other, radicals, such as the “McCoy” radical, with 0-band decompositions of semigroups. A radical \(\rho\) is “restorable by the components of every 0-band” if whenever S is a 0-band \(\{S_{\alpha}:\) \(\alpha\in \Omega \}\), (where the components \(S_{\alpha}\) are 0-disjoint), then \(\rho\) (S) is the largest ideal contained in \(\cup \{\rho (S_{\alpha}):\) \(\alpha\in \Omega \}\). Amongst other results, it is shown that the two radicals above have this property.
Reviewer: P.R.Jones

MSC:

20M11 Radical theory for semigroups
20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
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References:

[1] Adams D.H.,Semigroups with no nonzero nilpotent elements, Math. Z., 123 (1971), 168–176. · Zbl 0219.20041 · doi:10.1007/BF01110115
[2] Bosak J.,On radicals of semigroups, Mat.-fyz. časopis, 12 (1962), 230–234. · Zbl 0214.03506
[3] Bosak J.,On radicals of semigroups, Mat. časopis, 18 (1968), 204–212. · Zbl 0169.33301
[4] Chick H., B.J. Gardner,The preservation of some ring properties by semilattice sums, Commun. Algebra, 15 (1987), 1017–1038. · Zbl 0613.16008 · doi:10.1080/00927878708823455
[5] Clifford A., G. Preston,The algebraic theory of semigroups, 1964. · Zbl 0111.03403
[6] Eqbal Ahmed, R. Wiegandt,On lower radicals of semigroups, Math. Nachr., 57 (1973), 163–167. · Zbl 0265.20049 · doi:10.1002/mana.19730570109
[7] Fountain J., V. Gould,Completely O-simple semigroups of quotients. II, Contributions to General Algebra, Wien, 1985, 115–124. · Zbl 0574.20047
[8] Gardner B.J.,Radicals of supplementary semilattice sums of associative rings, Pacif. J. Math., 58 (1975), 387–392. · Zbl 0281.16007
[9] Grigor R.S.,On semigroups without nilpotent elements, Abstracts of I All-Union simposium on semigroup theory, Sverdlovsk, 1969, p.24 (Russian).
[10] Grigor R.S.,On the theory of radicals of semigroups. I, Mat. Issled. (Kishiniov), 6 (1971), 37–55 (Russian).
[11] Grigor R.S.,On the theory of radicals of semigroups. II, Mat. Issled. (Kishiniov), 8 (1973), 28–46 (Russian).
[12] Hoffman A.E.,Radicals of semigroups with zero, Notices Amer. Math. Soc., 20 (1973), A 359.
[13] Joulain C.,Sur les anneaux non commutatifs. I.Radical, Semin. P. Dubreil, 15 (1963), 1–13. · Zbl 0122.28801
[14] Kelarev A.V.,Radicals and O-bands of semigroups, preprint (Russian). · Zbl 0684.20051
[15] Kelarev A.V.,Prevarieties, radicals and bands of associative rings, XVIII All-Union algebraic conference, Kishiniov, 1985, part I, 242 (Russian).
[16] Kelarev A.V.,Radicals of semigroup rings of commutative semigroups, XIX All-Union algebraic conference, Lvov, 1987, part II, 122.
[17] Kelarev A.V.,Radicals and bands of semigroups, Isv. vysh. uch. sav., Matematika (to appear, in Russian).
[18] Kelarev A.V.,Hereditary radicals and bands of associative rings (to appear, in Russian). · Zbl 0756.16010
[19] Kelarev A.V.,A description of the radicals of semigroup algebras of commutative semigroups (to appear, in Russian). · Zbl 0851.20058
[20] Kozhevnikov O.B.,On a generalization of the concept of complete regularity, Associativnie dejstvija, 1983, 50–56 (Russian).
[21] Lallement G., M. Petrich,Some remarks concerning completely O-simple semigroups, Bull. Amer. Math. Soc., 70 (1964), 777–778. · Zbl 0126.04002 · doi:10.1090/S0002-9904-1964-11235-0
[22] Lallement G., M. Petrich,Decompositions I-matricielles d’un demi-grouppe, J. Math. Pures Appl., 45 (1966), 67–117.
[23] Lallement G., M. Petrich,A generalization of the Rees theorem in semigroups, Acta Sci. Math., 30 (1969), 113–132. · Zbl 0205.01703
[24] Leeuwen L.C.A. van, C. Roos, R. Wiegandt,Characterizations of semisimple classes, J. Austral. Math. Soc., 23 (1977), 172–182. · Zbl 0356.16003 · doi:10.1017/S1446788700018176
[25] Luh J.On the concept of radicals of semigroup having kernel, Portugaliae Math., 19 (1960), 189–198. · Zbl 0095.01405
[26] Marki L.,A note on quasi-ideals and O-matrix decomposition of semigroups with zero, preprint. · Zbl 0408.20049
[27] Marki L., P.N. Stewart, R. Wiegandt,Radicals and decomposability of semigroups and rings, Annales Univ. Sci. Budapest. Sect. Math., 18 (1975), 27–36. · Zbl 0331.16007
[28] Marki L.,Structure theorems on certain regular and inverse semigroups, Czechoslovak Math. J., 27 (1977), 388–393. · Zbl 0381.20045
[29] Marki L.,Radical semisimple classes and varieties of semigroups with zero, Algebraic theory of semigroups, Amsterdam, 1979, 357–369. · Zbl 0408.20049
[30] Shevrin L.N.,On the general semigroup theory, Mat. Sbornik, 59 (1961), 367–386 (Russian).
[31] Shevrin L.N.,On locally finite semigroups, Sov. Math. Dokl., 162 (1965), 770–773 (Russian). · Zbl 0144.01105
[32] Steinfeld O.,On semigroups which are unions of completely O-simple subsemigroups, Czehoslovak. Math. J., 16 (1966), 63–69. · Zbl 0141.02002
[33] Šulka R.,On nilpotent elements, ideals and radicals of a semigroup, Mat.-fys. časopis, 13 (1963), 209–221. · Zbl 0131.01902
[34] Šulka R.,Note on the Sevrin radical in semigroups, Mat. časopis, 18 (1968), 57–58. · Zbl 0174.04402
[35] Weissglass J.,Semigroup rings and semilattice sums of rings, Proc. Amer. Math. Soc., 39 (1973), 471–478. · Zbl 0272.16008 · doi:10.1090/S0002-9939-1973-0322092-4
[36] Wiegandt R.,On the structure of lower radical semigroups, Czechoslovak Math. J., 22 (1972), 1–6. · Zbl 0238.20081
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