The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner products, where $k$ and $s$ are positive integers and $A$ is an abelian group of odd order. Based on the well-know characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length $2^s$ over some Galois extensions of the ring $\mathbb{F}_{2^k}+u\mathbb{F}_{2^k}$, where $u^2=0$. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of length $p^s$ over $\mathbb{F}_{p^k}+u\mathbb{F}_{p^k}$ are given. Combining these results, the complete enumeration of self-dual abelian codes in $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ is therefore obtained.
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Leibniz algebras. We study Leibniz algebras containing abelian subalgebras of codimension 1, solvable and supersolvable Leibniz algebras for codimensions 1 and 2, nilpotent Leibniz algebras in case of codimension 2 and we also analyze the case of k-abelian p-filiform Leibniz algebras. Throughout the paper, we also give examples to clarify some results and the need for restrictions on the underlying field.
Let R be a commutative ring with unity $1\in R$. In this article, we introduce the concept of prime principal right ideal rings (\textbf{PPRIR}), A prime ideal P of R is said to be prime principal right ideal (\textbf{PPRI}) is given by $P =\{ ar : r\in R\}$ for some element a. The ring R is said to be prime principal right ideal ring (\textbf{PPRIR}) if every prime ideal of R is a prime principal right ideal (\textbf{PPRI}). A prime principal right ideal ring R is called a prime principal right ideal domain (\textbf{PPRID}) if R is a domain. Several properties and characteristics of prime pri…
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension $1$ and supersolvable Zinbiel algebras in which such subalgebras have codimension $2$, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for $α$ and $β$ for the low dimensional Zinbiel algebras over the complex field that have been classified.
Let $R$ be a commutative $G$-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal $I$ of $R$ is said to be graded radically principal if $Grad(I)=Grad(\langle c\rangle)$ for some homogeneous $c\in R$, where $Grad(I)$ is the graded radical of $I$. The graded ring $R$ is said to be graded radically principal if every graded ideal of $R$ is graded radically principal. We study graded radically principal rings. We prove an analogue of the Cohen theorem, in the graded case, precisely, a graded ring is graded radically princ…
We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.
For any constacyclic code over a finite commutative chain ring of length coprime to the characteristic of the ring, we construct explicitly generator polynomials and check polynomials, and exhibit a BCH bound for such constacyclic codes. As a consequence, such constacyclic codes are principal. Further, we get a necessary and sufficient condition that the cyclic codes over a finite commutative principal ideal ring are all principal. This condition is still sufficient for constacyclic codes over such rings being principal.
