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A family of noetherian rings with their finite length modules under control

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

We investigate the category mod Λ of finite length modules over the ring Λ = A ⊗ k Σ , where Σ is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k -algebra. Each simple module E j gives rise to a quasiprogenerator P j = A ⊗ E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that mod Λ ≃ ∐ j b a d h b o x P j . As a consequence we can (1) construct for each elementary k -algebra A over a finite field k a nonartinian noetherian ring Λ such that mod A ≃ mod Λ , (2) find twisted...

A generalization of the Auslander transpose and the generalized Gorenstein dimension

Yuxian Geng (2013)

Czechoslovak Mathematical Journal

Let R be a left and right Noetherian ring and C a semidualizing R -bimodule. We introduce a transpose Tr c M of an R -module M with respect to C which unifies the Auslander transpose and Huang’s transpose, see Z. Y. Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791–5812, in the two-sided Noetherian setting, and use Tr c M to develop further the generalized Gorenstein dimension with respect to C . Especially, we generalize the Auslander-Bridger formula to the generalized...

Actions of Hopf algebras on pro-semisimple noetherian algebras and their invariants

Andrzej Tyc (2001)

Colloquium Mathematicae

Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X₁,...,Xₙ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants A H is a noetherian Cohen-Macaulay...

Division dans l'anneau des séries formelles à croissance contrôlée. Applications

Augustin Mouze (2001)

Studia Mathematica

We consider subrings A of the ring of formal power series. They are defined by growth conditions on coefficients such as, for instance, Gevrey conditions. We prove a Weierstrass-Hironaka division theorem for such subrings. Moreover, given an ideal ℐ of A and a series f in A we prove the existence in A of a unique remainder r modulo ℐ. As a consequence, we get a new proof of the noetherianity of A.

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