Algebra and Geometry

This volume contains five review articles, three in the Al- gebra part and two in the Geometry part, surveying the fields of ring theory, modules, and lattice theory in the former, and those of integral geometry and differential-geometric methods in the calculus of variations in the latter. The literature covered is primarily that published in 1965-1968. v CONTENTS ALGEBRA RING THEORY L. A. Bokut', K. A. Zhevlakov, and E. N. Kuz'min 1. Associative Rings. . . . . . . . . . . . . . . . . . . . 3 2. Lie Algebras and Their Generalizations. . . . . . . 13 ~ 3. Alternative and Jordan Rings. . . . . . . . . . . . . . . . 18 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 MODULES A. V. Mikhalev and L. A. Skornyakov 1. Radicals. . . . . . . . . . . . . . . . . . . 59 2. Projection, Injection, etc. . . . . . . . . . . . . . . . . . . 62 3. Homological Classification of Rings. . . . . . . . . . . . 66 4. Quasi-Frobenius Rings and Their Generalizations. . 71 5. Some Aspects of Homological Algebra . . . . . . . . . . 75 6. Endomorphism Rings . . . . . . . . . . . . . . . . . . . . . 83 7. Other Aspects. . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 91 LATTICE THEORY M. M. Glukhov, 1. V. Stelletskii, and T. S. Fofanova 1. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . ' 111 2. Identity and Defining Relations in Lattices . . . . . . 120 3. Distributive Lattices. . . . . . . . . . . . . . . . . . . . . 122 vii viii CONTENTS 4. Geometrical Aspects and the Related Investigations. . . . . . . . . . . . * . . * . . . . . . . . . * 125 5. Homological Aspects. . . . . . . . . . . . . . . . . . . . . . 129 6. Lattices ofCongruences and of Ideals of a Lattice . . 133 7. Lattices of Subsets, of Subalgebras, etc. . . . . . . . . 134 8. Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 136 9. Topological Aspects. . . . . . . . . . . . . . . . . . . . . . 137 10. Partially-Ordered Sets. . . . . . . . . . . . . . . . . . . . 141 11. Other Questions. . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 GEOMETRY INTEGRAL GEOMETRY G. 1. Drinfel'd Preface . . . . . . . . .