In this paper, we study the generators of the module of derivations of the ring of invariants for some dihedral groups. Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text] and [Formula: see text] be a finite dihedral group. Let [Formula: see text] be the ring of invariants obtained by the linear action of [Formula: see text] on [Formula: see text]. In [ R. V. Gurjar and A. Patra, On minimum number of generators for some derivation modules, J. Pure Appl. Algebra [Formula: see text](11) (2022) 107105], Gurjar–Patra proved that [Formula: see text]. We will give a better bound on [Formula: see text] in this paper.

In this paper, Lie [Formula: see text]-isomorphisms on nest algebras and von Neumann algebras are constructed, which can be viewed as a common generalization of Lie isomorphisms and Lie triple isomorphisms. Besides, several 2-local Lie [Formula: see text]-isomorphisms and 2-local Lie [Formula: see text]-∗-isomorphisms on certain von Neumann algebras are studied.

We introduce the notion of cosymplectic structure on Jacobi–Jordan algebras, and we state that they are related to symplectic Jacobi–Jordan algebras. We show in particular, that they support a right-skew-symmetric product. We also study the double extension constructions of cosymplectic Jacobi–Jordan algebras and give a complete classification in dimension five.

In this paper, we first study when two-dimensional cyclic codes of length [Formula: see text] have generating sets of a certain form. Then we deduce that several of the main results of the papers [Z. Rajabi and K. Khashyarmanesh, Repeated-root two-dimensional constacyclic codes of length [Formula: see text], Finite Fields Appl. 50 (2018) 122–137] and [Z. Sepasdar and K. Khashyarmanesh, Characterizations of some two-dimensional cyclic codes correspond to the ideals of [Formula: see text], Finite Fields Appl. 41 (2016) 97–112] are not accurate. We also present counterexamples for these results and find other conditions needed for them to hold.

We investigate Gorenstein projective, injective, flat modules and dimensions over a polynomial ring using the equivalent category of the module category over the polynomial ring. Several estimates of Gorenstein homological dimensions of modules over the polynomial ring are obtained.

The category [Formula: see text] of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory [Formula: see text] of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers.

We investigate the relative lie algebroid connections on a holomorphic vector bundle over a family of compact complex manifolds (or smooth projective varieties over [Formula: see text]). We provide a sufficient condition for the existence of a relative Lie algebroid connection on a holomorphic vector bundle over a complex analytic family of compact complex manifolds. We show that the relative Lie algebroid Chern classes of a holomorphic vector bundle admitting relative Lie algebroid connection vanish, if each of the fibers of the complex analytic family is compact and Kähler. Moreover, we consider the moduli space of relative Lie algebroid connections and we show that there exists a natural relative compactification of this moduli space.

We prove that the Brinkmann Problems ( BrP & BrCP ) and the Twisted-Conjugacy Problem ( TCP ) are decidable for any endomorphism of a free-abelian times free (FATF) group [Formula: see text]. Furthermore, we prove the decidability of the two-sided Brinkmann Conjugacy Problem ( 2BrCP ) for monomorphisms of FATF groups (and combine it with TCP ) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.