Decidability questions for theories of modules.

*(English)*Zbl 0795.03016
Oikkonen, Juha Markku Robert (ed.) et al., Logic colloquium ’90. ASL summer meeting in Helsinki, Finland from July 15 to July 22, 1990. Berlin: Springer-Verlag. Lect. Notes Log. 2, 266-280 (1993).

S. Burris, R. McKenzie, M. Valeriote and R. Willard classified the decidable locally finite varieties up to the classification of the decidable locally finite affine varieties. S. Burris and R. McKenzie showed that this later problem was equivalent to classifying the finite rings with identity which have a decidable theory of unitary left modules.

First we briefly review the work of the four mentioned authors and then we concentrate on the decidability question for theories of modules. The main themes are the following: what are the relationships

– between the representation type of a finite-dimensional \(K\)-algebra and the decidability of its theory of left-modules,

– between theories of modules over rings which are Morita equivalent,

– between the theory of \(R\)-modules and the theory of finitely presented \(R\)-modules.

As an illustration of the first point, we present a partial classification of commutative Artinian rings with decidable theory of modules.

For the entire collection see [Zbl 0782.00083].

First we briefly review the work of the four mentioned authors and then we concentrate on the decidability question for theories of modules. The main themes are the following: what are the relationships

– between the representation type of a finite-dimensional \(K\)-algebra and the decidability of its theory of left-modules,

– between theories of modules over rings which are Morita equivalent,

– between the theory of \(R\)-modules and the theory of finitely presented \(R\)-modules.

As an illustration of the first point, we present a partial classification of commutative Artinian rings with decidable theory of modules.

For the entire collection see [Zbl 0782.00083].

Reviewer: F.Point

##### MSC:

03B25 | Decidability of theories and sets of sentences |

03C60 | Model-theoretic algebra |

13C05 | Structure, classification theorems for modules and ideals in commutative rings |

08C10 | Axiomatic model classes |