By L. A. Bokut’, K. A. Zhevlakov, E. N. Kuz’min (auth.), R. V. Gamkrelidze (eds.)

This quantity includes 5 assessment articles, 3 within the Al gebra half and within the Geometry half, surveying the fields of ring conception, modules, and lattice thought within the former, and people of critical geometry and differential-geometric tools within the calculus of diversifications within the latter. The literature lined is essentially that released in 1965-1968. v CONTENTS ALGEBRA RING thought L. A. Bokut', ok. A. Zhevlakov, and E. N. Kuz'min § 1. Associative jewelry. . . . . . . . . . . . . . . . . . . . three § 2. Lie Algebras and Their Generalizations. . . . . . . thirteen ~ three. replacement and Jordan jewelry. . . . . . . . . . . . . . . . 18 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 MODULES A. V. Mikhalev and L. A. Skornyakov § 1. Radicals. . . . . . . . . . . . . . . . . . . fifty nine § 2. Projection, Injection, and so forth. . . . . . . . . . . . . . . . . . . sixty two § three. Homological class of earrings. . . . . . . . . . . . sixty six § four. Quasi-Frobenius jewelry and Their Generalizations. . seventy one § five. a few elements of Homological Algebra . . . . . . . . . . seventy five § 6. Endomorphism jewelry . . . . . . . . . . . . . . . . . . . . . eighty three § 7. different features. . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , ninety one LATTICE concept M. M. Glukhov, 1. V. Stelletskii, and T. S. Fofanova § 1. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . " 111 § 2. identification and Defining family in Lattices . . . . . . one hundred twenty § three. Distributive Lattices. . . . . . . . . . . . . . . . . . . . . 122 vii viii CONTENTS § four. Geometrical points and the comparable Investigations. . . . . . . . . . . . • . . • . . . . . . . . . • a hundred twenty five § five. Homological features. . . . . . . . . . . . . . . . . . . . . . 129 § 6. Lattices of Congruences and of beliefs of a Lattice . . 133 § 7. Lattices of Subsets, of Subalgebras, and so forth. . . . . . . . . 134 § eight. Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 136 § nine. Topological points. . . . . . . . . . . . . . . . . . . . . . 137 § 10. Partially-Ordered units. . . . . . . . . . . . . . . . . . . . 141 § eleven. different Questions. . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 GEOMETRY crucial GEOMETRY G. 1. Drinfel'd Preface . . . . . . . . .

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Amer. Math. , 124(1):77-93 (1966). 361. D. B. Ion, Asupra algebrelor Jordan de tip A. Studii si cercetari mat. Acad. RPR, 17(2):301-310 (1965). 362. D. B. Ion, Asupra unor proprietli~i ale constantelor de structura ale algebrelor Jordan de tip A. Studii ~i cercetari mat. Acad. RPR, 19(7):1031-1038 (1967). 363. D. B. Ion, Asupra unor propriediti ale radacinilor algebrelor Jordan de tip A. Studii >si cercetari mat. Acad. RPR, 18(2):309-313 (1966). 364. 1. D. Ion, Radicalul limitei projective de inele asociative.

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Kawada, On Kothe's problem concerning algebras for which every indecomposable module is cyclic. Ill. Sci. Repts. Tokyo Kyoiku Daigaku A, 8(196-201): 1-250 (1965). 399. S. M. Kaye, Ring theoretic properties of matrix rings. Canad. Math. , 10(3):365-374 (1967). RING THEORY 43 400. O. H. Kegel, On rings that are sums of two subrings. J. Algebra, 1(2):103-109 (1964). 401. A. Kertesz, On the existence of a left unit element in a noetherian or in an artinian ring. Bull. Acad. polon. sci. Sec. sci. , astron.