We prove a differential analogue of Hilbert’s irreducibility theorem. Let L be a linear differential operator with coefficients in C ( X ) ( x ) that is irreducible over C ( X ) ¯ ( x ) , where X is an irreducible affine algebraic variety over an algebraically closed field C of characteristic zero. We show that the set of c ∈ X ( C ) such that the specialized operator L c of L remains irreducible over C ( x ) is Zariski dense in X ( C ) .

Let p 1, p 2 be two distinct prime integers, let n be a positive integer, n ≥ 3 and let ξn be a primitive root of order n of the unity. In the 3rd section of this paper we obtain a complete characterization for a quaternion algebra H ( p 1 , p 2 ) to be a division algebra over the nth cyclotomic field Q ( ξ n ) , when n ∈ { 3 , 4 , 6 , 7 , 8 , 9 , 11 , 12 } and we also obtain a characterization for a quaternion algebra H ( p 1 , p 2 ) to be a division algebra over the nth cyclotomic field Q ( ξ n ) , when n ∈ { 5 , 10 } . In the 4th section we obtain a complete characterization for a quaternion algebra H Q ( ξ n ) ( p 1 , p 2 ) to be a division algebra, when n = l k , with l a prime integer, l ≡ 3 (mod 4) and k a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra H Q ( ξ l ) ( p 1 , p 2 ) to be a division algebra, when l is a Fermat prime number.

Let P O P D I n ( P O D I n ) be the semigroup consisting of all injective orientation-preserving (order-preserving) and order-decreasing transformations on the finite chain { 1 < ⋯ < n } . For 1 ≤ r ≤ n , let P O D I ( n , r ) = { α ∈ P O D I n : | im ( α ) | ≤ r } and P O P D I ( n , r ) = { α ∈ P O P D I n : | im ( α ) | ≤ r } . In this paper, we find the minimal generating sets, and so the ranks of P O D I ( n , r ) and P O P D I ( n , r ) . We also consider the nilpotent subsemigroup of P O D I ( n , r ) .

Let w ↦ ( P ( w ) , Q ( w ) ) be the Robinson-Schensted correspondence between the symmetric group S n and the set of pairs of standard tableaux with the same shapes. We show that each Kazhdan-Lusztig basis (KL basis for short) element C ′ w can be expressed as a linear combination of some f s t which satisfies that s ¥ P ( w ) ∗ , t ¥ Q ( w ) ∗ , where “ ¥ ” is the dominance (partial) order between standard tableaux, u ∗ denotes the conjugate of u for each standard tableau u , { f s t | s , t ∈ Std ( λ ) , λ⊢n } is the seminormal basis of the Iwahori-Hecke algebra associated to S n . As a result, we generalize an earlier result of Geck on the relation between the KL basis and the Murphy basis. Similar relations between the twisted KL basis, the dual seminormal basis and the dual Murphy basis are obtained.

Using techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman et al. under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate GRH, for any (rational) prime p the set E ( p ) = { q prime | X q − 1 factors in F p [ X ; M ] } , where M = 〈 2 , 3 〉 = N 0 ∖ { 1 } , contains a subset with positive natural density. In particular E ( p ) ≠ ∅ . This proves that M is not a so-called “Matsuda monoid” of any positive type. For p = 2 , 3 this was observed by Kulosman, who provided factorizations of X 7 − 1 and X 11 − 1 in F 2 [ X ; M ] and F 3 [ X ; M ] , respectively. Our results explain and reproduce both of these factorizations, as well.

We consider a problem proposed by Oliynyk who gave a sufficient condition for block Toeplitz matrices to generate a free product of p-groups. In this work we present the necessary condition. Namely, we prove that the representants of the mentioned block Toeplitz matrices should not be simultaneously triangularizable.

A ring R with center C is said to be centrally essential if the module R C is an essential extension of the module C C . In this paper, we study properties of ideals of centrally essential rings, centrally essential quaternion algebras, and group rings of Hamiltonian groups.

We consider the category of partial actions, where the group and the set upon which the group acts can vary. Within this framework, we develop a theory of quotient partial actions and prove that this category is both (co)complete and encompasses the category of groupoids as a full subcategory. In particular, we establish the existence of a pair of adjoint functors, denoted as Φ : Grpd → PA and Ψ : PA → Grpd , with the property that Ψ Φ ≅ 1 Grpd . Next, for a given groupoid Γ, we provide a characterization of all partial actions that allow the recovery of the groupoid Γ through Ψ . This characterization is expressed in terms of certain normal subgroups of a universal group constructed from Γ .