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Products of small modules

Peter Kálnai, Jan Žemlička (2014)

Commentationes Mathematicae Universitatis Carolinae

Module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal.

Representation-directed algebras form an open scheme

Stanislaw Kasjan (2002)

Colloquium Mathematicae

We apply van den Dries's test to the class of algebras (over algebraically closed fields) which are not representation-directed and prove that this class is axiomatizable by a positive quantifier-free formula. It follows that the representation-directed algebras form an open ℤ-scheme.

Representation-finite triangular algebras form an open scheme

Stanisław Kasjan (2003)

Open Mathematics

Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.

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