## Continuous and discrete modules.(English)Zbl 0701.16001

London Mathematical Society Lecture Note Series, 147. Cambridge etc.: Cambridge University Press. 126 p. £12.50; \$ 22.95 (1990).
When von Neumann introduced continuous geometries (1936), he used the adjective “continuous” to emphasize the fact that these “pointless geometries” support dimension functions with a continuum of values (e.g., the intervals [0,1] or $$[0,\infty))$$. He also used “continuity” to label one of the key lattice-theoretic axioms of these geometries. In current terminology, a continuous lattice is a complete lattice L in which $$x\wedge (\bigvee y_ j)=\bigvee (x\wedge y_ j)$$ (upper continuity) and $$x\vee (\bigwedge y_ j)=\bigwedge (x\vee y_ j)$$ (lower continuity) for all $$x\in L$$ and all linearly ordered subsets $$\{y_ j\}\subseteq L$$. (With respect to a suitable order topology on L, upper and lower continuity say that the functions $$y\mapsto x\wedge y$$ and $$y\mapsto x\vee y$$ are continuous.) Taking account of von Neumann’s coordinatization theorem - which says that apart from a few exceptions, any complemented modular lattice is isomorphic to the lattice of principal right ideals of a regular ring - a regular ring is called (right) continuous provided its lattice of principal right ideals is (upper) continuous.
Utumi (1960) proved that a regular ring is right continuous if and only if every right ideal is essential in a principal right ideal. In 1965, he showed that the direct sum decomposition theory in such a ring also holds for rings in which (a) every right ideal is essential in a direct summand, and (b) any right ideal isomorphic to a direct summand is itself a direct summand; these rings he called “right continuous rings”. This concept was transferred to modules, and generalized to e.g. quasi- continuity and $$(quasi$$-)$$\aleph_ 0$$-continuity, by various authors, such as Jeremy (1971), Mohamed and Bouhy (1977), and Kutami (1983). The starting point of the present treatise is the concept of quasi- continuity: a module M is quasi-continuous provided (a) every submodule of M is essential in a direct summand of M, and (b) whenever $$M_ 1$$ and $$M_ 2$$ are direct summands of M with $$M_ 1\cap M_ 2=0$$, then $$M_ 1\oplus M_ 2$$ is a direct summand.
The monograph under review presents a complete and current account of the subject of continuous and quasi-continuous modules, and of modules satisfying dual conditions, which the authors baptize “quasi-discrete”. The main theme of the book is the development and exploitation of the generous supply of direct summands in such modules; we mention some sample results. (1) A module is quasi-continuous if and only if it inherits all direct sum decompositions from its injective hull. (2) Two direct summands of a quasi-continuous module are isomorphic if and only if their injective hulls are isomorphic. (3) Any quasi-continuous module is a direct sum of a quasi-injective module and a module with no submodules of the form $$X\oplus X$$. (4) In a quasi-continuous module, isomorphic directly finite summands have isomorphic complements; moreover, directly finite continuous modules cancel from external direct sums. (5) All continuous modules satisfy the exchange property.
In an appendix, various related topics - such as supplemented modules - are discussed, a historical overview of von Neumann $$(W^*$$-)algebras and continuous geometries is presented, and a sketch of recent work on $$\aleph_ 0$$-continuity is given. The book concludes with a list of 24 open problems, and an extensive bibliography.
Reviewer: K.R.Goodearl

### MSC:

 16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras 16D10 General module theory in associative algebras 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 16D40 Free, projective, and flat modules and ideals in associative algebras 16D50 Injective modules, self-injective associative rings 06B35 Continuous lattices and posets, applications 46L10 General theory of von Neumann algebras