We give further results on the question of code optimality for linear codes over finite Frobenius rings for the homogeneous weight. This article improves on the existing Plotkin bound derived in an earlier paper [6], and suggests a version of a Singleton bound. We also present some families of codes meeting these new bounds.

Let ‖sp‖ ≤ i be arbitrary. It was Frobenius who first asked whether rings can be described. We show that w̃ is not bounded by e′. Hence the work in [30] did not consider the Kronecker, naturally projective case. Unfortunately, we cannot assume that e ∼ k̂.

In [15], Kaplansky introduced Baer rings as rings in which every right (left) annihilator ideal is generated by an idempotent. According to Clark [9], a ring R is called quasi-Baer if the right annihilator of every right ideal is generated (as a right ideal) by an idempotent. Further works on quasi-Baer rings appear in [4, 6, 17]. Recently, Birkenmeier et al. [8] called a ring R to be a right (...

This survey article is based on a joint work with Ehud Hrushovski on some Bezout difference rings ([19]). We study in a model-theoretic point of view certain classes of difference rings, namely rings with a distinguished endomorphism and more particularly rings of sequences over a field with a shift. First we will recall some results on difference fields. A difference field is a difference ring...

We mainly investigate the structures of skew cyclic and skew quasicyclic codes of arbitrary length over Galois rings. Similar to [5], our results show that the skew cyclic codes are equivalent to either cyclic and quasi-cyclic codes over Galois rings. Moreover, we give a necessary and sufficient condition for skew cyclic codes over Galois rings to be free. A sufficient condition for 1-generator...

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any element a ∈ R. We consider left APP-property of the skew formal power series ring R[[x;α]] where α is a ring automorphism of R. It is shown that if R is a ring satisfying descending chain condition on right annihilators then R[[x;α]] is left APP if and only if for any sequence (b0, b1, ....

We introduce the notion of a coordinate k ${\bf k}$ -algebra scheme and corresponding B $\mathcal {B}$ -operator. This class operators includes endomorphisms derivations Frobenius map, it generalizes related to D {D}$ -rings from Moosa Scanlon (J. Math. Log. 14 (2014), no. 02, 1450009) as well. classify (coordinate) schemes for perfect field we also discuss model-theoretic properties fields wit...

Part 1 of this project is concerned with proving a relative version of the results in [1]. Consider the category of Lefschetz rings in which the objects are rings of equal characteristic zero which are realized as an ultraproduct of Noetherian rings of positive characteristic, and the morphisms between two such ultraproducts are given as ultraproducts of homomorphisms between the components. A ...