Continuous and Discrete Modules
Cambridge University Press, Feb 22, 1990 - Mathematics - 126 pages
Continuous and discrete modules are, essentially, generalizations of infective and projective modules respectively. Continuous modules provide an appropriate setting for decomposition theory of von Neumann algebras and have important applications to C*-algebras. Discrete modules constitute a dual concept and are related to number theory and algebraic geometry: they possess perfect decomposition properties. The advantage of both types of module is that the Krull-Schmidt theorem can be applied, in part, to them. The authors present here a complete account of the subject and at the same time give a unified picture of the theory. The treatment is essentially self-contained, with background facts being summarized in the first chapter. This book will be useful therefore either to individuals beginning research, or the more experienced worker in algebra and representation theory.
Injectivity and related concepts
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A—injective abelian groups Amer arbitrary artinian assume attached maximal ideal cancellation property commutative noetherian ring complements summands consequently contains continuous geometries continuous modules contradiction cyclic modules decomposition Dedekind domains deﬁned Deﬁnition direct sum directly ﬁnite endomorphism ring epimorphism exists extending modules finite ﬁnite exchange property ﬁnite subset ﬁrst following are equivalent Goodearl hence hollow discrete modules hollow modules homomorphism idempotents implies injective hull injective modules internal cancellation property isomorphism Jain lifting property lsTn M1 Q M2 Math maximal ideal Miiller Mohamed monomorphism noetherian rings non—zero orthogonal idempotents Osaka Paciﬁc prime ideal projective modules PROOF Proposition purely inﬁnite quasi—continuous modules quasi—discrete module quasi—injective modules quasi—projective module quotient ﬁeld R—module regular rings relative projectivity relatively injective right continuous satisﬁes self—injective square free sum of hollow sum of indecomposable superspectivity Theorem torsion uniform modules unique Univ Utumi valuation ring von Neumann algebras Zorn’s Lemma Zoschinger