Journal of Algebra and Its Applications

An n-ary endofunction on an abelian group G is a function f : Gⁿ → G such that f(θg₁,…,θg_n) = θ f(g₁,…,g_n) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: for some collection of integers {l_i : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form for some collection of central endomorphisms {λ_i : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

In this note we consider finite translation planes with large translation complements. In particular, we characterize finite affine Desarguesian translation planes in two ways, according to the existence of subgroups in the translation complement that are divisible by relatively large integers, together with modest additional restrictions.

A formula for computing the discriminant of any Abelian number field K is given. It is presented as a function of the conductor m of K and of the degrees of the fields K ∩ ℚ(ζ_p^α) over ℚ, where p runs through the set of primes that divide m, and p^α is the greatest power of p that divides m.

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [4] that only certain combinations of the algebraic parameters (field size, length, dimension, and Forney indices) can occur for such cyclic codes. We will investigate whether all these combinations can indeed be realized by a suitable cyclic code and, if so, how to construct such a code. A complete characterization and construction will be given for minimal cyclic codes. It is derived from a detailed investigation of the units in the skew-polynomial ring.

We determine the identities of degree ≤ 6 satisfied by the symmetric (Jordan) product a○b = ab + ba and the associator [a,b,c] = (ab)c - a(bc) in every nonassociative algebra. In addition to the commutative identity a○b = b○a we obtain one new identity in degree 4 and another new identity in degree 5. We demonstrate the existence of further new identities in degree 6. These identities define a variety of binary-ternary algebras which generalizes the variety of Jordan algebras in the same way that Akivis algebras generalize Lie algebras.

We observe that the Dieudonné determinant induces a non-negative degree function on the ring of matrices over a skew polynomial ring. We then apply this degree function to two examples. In the first one, we find an expression for the rank of the kernel of an algebraic endomorphism of over a field of characteristic p > 0. In the second, we calculate the dimension of the solution space of linear matrix differential equations.

Let X be an abelian variety defined over an algebraically closed field k of arbitrary characteristic. The aim of this note is to show that the scheme , which parametrizes germs of formal curves on X, is the Picard scheme of a formal scheme.

In this work we continue the discussion of the conatural classes introduced in [1]. We prove that the collection R-conat of all conatural classes of left modules over a ring R is a set, and it is a boolean lattice. Afterwards we study relationships between some lattices of module classes: R-conat and R-tors, R-her and R-quot. As a consequence of the developed theory we obtain characterizations of left MAX rings and artinian principal ideal rings.