367 research outputs found
Ideal codes over separable ring extensions
This paper investigates the application of the theoretical algebraic notion
of a separable ring extension, in the realm of cyclic convolutional codes or,
more generally, ideal codes. We work under very mild conditions, that cover all
previously known as well as new non trivial examples. It is proved that ideal
codes are direct summands as left ideals of the underlying non-commutative
algebra, in analogy with cyclic block codes. This implies, in particular, that
they are generated by an idempotent element. Hence, by using a suitable
separability element, we design an efficient algorithm for computing one of
such idempotents
Calabi-Yau coalgebras
We provide a construction of minimal injective resolutions of simple
comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau
condition of algebras, we introduce the Calabi-Yau condition to coalgebras.
Then we give some descriptions of Calabi-Yau coalgebras with lower global
dimensions. An appendix is included for listing some properties of cohom
functors
Dualities of artinian coalgebras with applications to noetherian complete algebras
A duality theorem of the bounded derived category of quasi-finite comodules
over an artinian coalgebra is established. Let be a noetherian complete
basic semiperfect algebra over an algebraically closed field, and be its
dual coalgebra. If is Artin-Schelter regular, then the local cohomology of
is isomorphic to a shift of twisted bimodule with
a coalgebra automorphism. This yields that the balanced dualinzing
complex of is a shift of the twisted bimodule . If
is an inner automorphism, then is Calabi-Yau
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