Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...

We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ∗ is a right (left) ideal if it satisfies L = LΣ∗ (L = Σ∗L). It is a two-sided ideal if L = Σ∗LΣ∗, and an all-sided ideal if L = Σ∗ L, the shuffle of Σ∗ with L. We prefer the term “quotient complexity” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient co...

Let $R$ be a regular local ring. $\mathbf {G}$ reductive $R$-group scheme. A conjecture of Grothendieck and Serre predicts that principal ${\mathbf {G}}$-bundle over is trivial if it the quotient field $R$. The known when contains field. We prove for large class rings

The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L′ are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L′ are the symmetric groups Sm and Sn of degrees m and n, respectively. Denote by ◦ any...