Totally categorical groups and rings

This paper is a contribution to applied categoricity theory. We try to classify isomorphism theorems in a certain area of algebra, by systematic use of the powerful model-theoretic methods of Morley, Shelah, and Ryll-Nardzewski. As far as we know, no results in applied categoricity were published before 1969. Since then the literature has grown quite rapidly. Before stating our main results, we give a survey of the previously known results. The main concepts under study have been K,,-categoricity, N,-categoricity, and w-stability. These concepts apply either to complete theories or to individual structures. Shelah’s notions of superstability and stability are receiving increasing attention, but only in the case of Abelian groups can we give them useful algebraic characterizations at present. In the case of &,-categoricity, satisfactory analyses have been given in the following cases:

[1]  Bruce I. Rose,et al.  Model theory of alternative rings , 1978, Notre Dame J. Formal Log..

[2]  A. Lachlan.  Two conjectures regarding the stability of ω-categorical theories , 1974 .

[3]  Б. И. Зильбер.  Группы и кольца, теория которых категорична , 1977 .

[4]  Angus Macintyre,et al.  ℵ0-Categoricity for rings without nilpotent elements and for boolean structures , 1976 .

[5]  B. A. F. Wehrfritz,et al.  Locally finite groups , 1973 .

[6]  G. Cherlin.  Model theoretic algebra: Selected topics , 1976 .

[7]  S. Feferman,et al.  The first order properties of products of algebraic systems , 1959 .

[8]  Irving Kaplansky,et al.  Fields and rings , 1969 .

[9]  John T. Baldwin,et al.  _{} is finite for ℵ₁-categorical , 1973 .

[10]  M. Hall.  The Theory Of Groups , 1959 .

[11]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[12]  John T. Baldwin,et al.  ℵ0-Categoricity and stability of rings , 1977 .

[13]  Gregory Cherlin.  Groups of small Morley rank , 1979 .

[14]  Walter Baur.  $\aleph_0$-Categorical Modules , 1975 .

[15]  Joseph G. Rosenstein.  ℵ0-Categoricity of groups , 1973 .

[16]  ℵ0 stable groups , 1978 .

[17]  John T. Baldwin,et al.  Logical stability in group theory , 1976, Journal of the Australian Mathematical Society.

[18]  Gerald E. Sacks,et al.  Saturated Model Theory , 1972 .

[19]  Angus Macintyre,et al.  On $ω_1$-categorical theories of abelian groups , 1971 .

[20]  Joseph G. Rosenstein,et al.  $ℵ_0$-categoricity of linear orderings , 1969 .

[21]  Alfred Tarski,et al.  Direct decompositions of finite algebraic systems , 1947 .

[22]  M. Morley.  Categoricity in power , 1965 .

[23]  Urlich Felgner.  $ℵ_1$ - Kategorische Theorien nicht-kommutativer Ringe , 1975 .

[24]  A. Macintyre.  On $ω_1$-categorical theories of fields , 1971 .

[25]  Paul C. Eklof,et al.  Model-completions and modules , 1971 .