We introduce the Singleton bounds for codes over a finite commutative quasi-Frobenius ring.

We introduce a notion of depth three tower of three rings C ⊆ B ⊆ A with depth two ring extension A |B recovered when B = C. If A = EndBC and B |C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension in [12, 13] such that if B |C is depth three, then A |C is depth two (a phenomenon of finite depth subfactors, see [20]). We provide a similar definition of f...

We introduce fusion bialgebras and their duals systematically study Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of rings. prove Hausdorff-Young inequality, uncertainty principles for duals. show that Schur product property, Young's inequality sum-set estimate hold bialgebras, but not always If ring is Grothendieck a catego...

the aim of this paper is to generalize the‎‎notion of pseudo-almost valuation domains to arbitrary‎ ‎commutative rings‎. ‎it is shown that the classes of chained rings‎ ‎and pseudo-valuation rings are properly contained in the class of‎ ‎pseudo-almost valuation rings; also the class of pseudo-almost‎ ‎valuation rings is properly contained in the class of quasi-local‎ ‎rings with linearly ordere...