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In this article, we commence to study the real (simple) left-symmetric algebras. From the known classification of certain complex (semi)simple left-symmetric algebras, we classify their corresponding real forms. We not only obtain the classification of real simple left-symmetric algebras in low dimensions, but also find certain examples of real simple left-symmetric algebras in higher dimensions. In particular, there exists a complex simple left-symmetric algebra without any real form. We also give a geometric construction for a class of real simple left-symmetric algebras. At last, we apply the classification results to study some structures related to geometry. [ABSTRACT FROM PUBLISHER]

In this article, Gorenstein FP-injective modules are introduced and investigated. A left R-module M is called Gorenstein FP-injective if there is an exact sequence … → E 1 → E 0 → E 0 → E 1 → … of FP-injective left R-modules with M = ker(E 0 → E 1) such that Hom R (P, −) leaves the sequence exact whenever P is a finitely presented left R-module with pd R (P) < ∞. Some properties of Gorenstein FP-injective modules are obtained. Several well-known classes of rings are characterized in terms of Gorenstein FP-injective modules. [ABSTRACT FROM PUBLISHER]

For a prime p, we denote by Bn the cyclic group of order pn. Let φ be a faithful irreducible character of Bn, where p is an odd prime. We study the p-group G containing Bn such that the induced character φG is also irreducible. The purpose of this article is to determine the subgroup NG(NG(Bn)) of G under the hypothesis [NG(Bn):Bn]4 ≦ pn. [ABSTRACT FROM AUTHOR]