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Acta Mathematica Hungarica
Authors: José Anquela, Teresa Cortés, Miguel Gómez-Lozano, and Mercedes Siles-Molina

Abstract  

We investigate the basic properties of the different socles that can be considered in not necessarily semiprime associative systems. Among other things, we show that the socle defined as the sum of minimal (or minimal and trivial) inner ideals is always an ideal. When trivial inner ideals are included, this inner socle contains the socles defined in terms of minimal left or right ideals.

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Abstract

For semiprime involution rings, we determine some ∗-minimal ∗-ideals using idempotent elements. Nevertheless, ∗-minimal ∗-biideals are characterized by idempotent elements. Moreover, the involutive version of a theorem due to Steinfeld, which investigates a semiprime involution ring A if A=SocA, is given. Finally, semiprime involution rings having no proper nonzero ∗-biideals are characterized.

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It is shown thatX is finite if and only ifC(X) has a finite Goldie dimension. More generally we observe that the Goldie dimension ofC(X) is equal to the Souslin number ofX. Essential ideals inC(X) are characterized via their corresponding z-filters and a topological criterion is given for recognizing essential ideals inC(X). It is proved that the Fréchet z-filter (cofinite z-filter) is the intersection of essential z-filters. The intersection of idealsO x wherex runs through nonisolated points inX is the socle ofC(X) if and only if every open set containing all nonisolated points is cofinite. Finally it is shown that if every essential ideal inC(X) is a z-ideal thenX is a P-space.

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.11.034 . [7] Chen , J. Ding , N. Yousif , M. F. 2004 On Noetherian rings with essential socle J. Aust. Math. Soc

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Acta Mathematica Hungarica
Authors: Daniel Ceretto, Esther García, and Miguel Gómez Lozano

. García , E. Gómez Lozano , M. 2004 The Jordan socle and finitary Lie algebras J. Algebra 280 635 – 654 10.1016/j.jalgebra.2004

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Nicholson, W. K. and Watters, J. F. , Rings with projective socle, Proc. Amer. Math. Soc. , 102 (1988), 443–450. Watters J. F. Rings with projective socle

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Ghaffari, A. and Medghalchi, A. R. , The socle and finite dimensionality of some Banach algebras, Proc. Indian Acad. Sci. , 115 (2005), 237–242. MR 2161736 ( 2006e :43002) Medghalchi A. R

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. Jantsky, B. 1979: Géologie du socle cristallin granitisé de la Montagne Mecsek. - Ann. Inst. Geol. Publ. Hung., 60, Budapest, 385 p. Géologie du socle cristallin granitisé de la Montagne Mecsek. Ann

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Central European Geology
Authors: Nawal Bouya, Hmidou El Ouardi, Hassania El Habibou, Véronique Ansan, and Eric Mercier

Hall London . E.B. Randrianasolo 2009 Nouvelles méthodes de cartographie sur le socle protérozoique du sud de Madagascar

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