##
**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.

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 |