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Rings over which all modules of a given type are almost categorical. (English. Russian original) Zbl 0584.03025
Algebra Logic 23, 113-124 (1984); translation from Algebra Logika 23, No. 2, 159-174 (1984).
The author’s characterization of almost categorical modules in the article reviewed above [see Zbl 0584.03024] is used for the study of rings with the condition of almost categoricity of all modules from one of the following classes: a) class of all R-modules, b) class of all injective R-modules, c) class of all projective R-modules (R is an associative ring with unity).
Theorem 1. The following conditions are equivalent: 1) All left R-modules are almost categorical. 2) All right R-modules are almost categorical. 3) The ring R is Artinian and semisimple.
Theorem 2. All injective left R-modules are almost categorical if and only if R is left Artinian.
Theorem 3. All projective left R-modules are almost categorical if and only if R is a left perfect and right coherent ring.
As corollaries, rings with the condition of categoricity of all modules from one of the above classes are characterized. It is also shown that if K is one of the classes b) or c), then the following conditions are equivalent: 1) K is categorical, 2) K is axiomatizable and complete, 3) K is axiomatizable and model complete.

MSC:
03C60 Model-theoretic algebra
08C10 Axiomatic model classes
16D40 Free, projective, and flat modules and ideals in associative algebras
03C35 Categoricity and completeness of theories
16D50 Injective modules, self-injective associative rings
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References:
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