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In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form \begin{document}$ [I_n \ | \ \Omega(v)], $\end{document} where \begin{document}$ I_n $\end{document} is the identity matrix and \begin{document}$ \Omega(v) $\end{document} is a composite matrix and search for binary self-dual codes with parameters \begin{document}$ [36,18, 6 \ \text{or} \ 8]. $\end{document} We next lift these codes over the ring \begin{document}$ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $\end{document} to obtain codes whose binary images are self-dual codes with parameters \begin{document}$ [72,36,12]. $\end{document} Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find \begin{document}$ 30 $\end{document} new Type I binary self-dual codes with parameters \begin{document}$ [72,36,12]. $\end{document}

Finding the minimum distance of linear codes is one of the main problems in coding theory. The importance of the minimum distance comes from its error-correcting and error-detecting capability of the handled codes. It was proven that this problem is an NP-hard that is the solution of this problem can be guessed and verified in polynomial time but no particular rule is followed to make the guess and some meta-heuristic approaches in the literature have been used to solve this problem. In this paper, swarm-based optimization techniques, bat and firefly, are applied to the minimum distance problem by integrating the algebraic operator to the handled algorithms.

We define additive skew G-codes over finite fields and discuss several dualities attached to these codes. We examine the properties of self-dual skew G-codes and in particular we show that the dual, under any duality, of an additive skew G-code is also an additive skew G-code. Additionally, we propose a matrix construction for additive skew G-codes and use it to construct several examples of extremal self-dual additive skew G-codes over the finite field $${\mathbb {F}}_4$$ . Such codes have a strong connection to quantum error correcting codes.

In this work, we study a new family of rings, ${\mathscr{B}}_{j,k}$ B j , k , whose base field is the finite field ${\mathbb {F}}_{p^{r}}$ F p r . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over ${\mathscr{B}}_{j,k}$ B j , k to a code over ${\mathscr{B}}_{l,m}$ B l , m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible $G^{2^{j+k}}$ G 2 j + k -code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group $${\mathbb {Z}}_{p^e}$$ . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with $$p \equiv 1 \pmod {4}$$ and e is odd with $$e>1$$ ; and all lengths that are $$0 \pmod {4}$$ when p is an odd prime with $$p \equiv 3 \pmod {4}$$ and e is odd with $$e>1.$$

We study the graded isoradical of a ring graded by a group. In particular, we compare the graded isoradical and the classical isoradical of a graded ring, examine the question of how the (graded) i...

In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the \begin{document}$ k^{th} $\end{document}-range neighbours, and search for binary \begin{document}$ [72, 36, 12] $\end{document} self-dual codes. In particular, we present six generator matrices of the form \begin{document}$ [I_{36} \ | \ \tau_6(v)], $\end{document} where \begin{document}$ I_{36} $\end{document} is the \begin{document}$ 36 \times 36 $\end{document} identity matrix, \begin{document}$ v $\end{document} is an element in the group matrix ring \begin{document}$ M_6(\mathbb{F}_2)G $\end{document} and \begin{document}$ G $\end{document} is a finite group of order 6, to which we employ the proposed algorithm and search for binary \begin{document}$ [72, 36, 12] $\end{document} self-dual codes directly over the finite field \begin{document}$ \mathbb{F}_2 $\end{document}. We construct 1471 new Type I binary \begin{document}$ [72, 36, 12] $\end{document} self-dual codes with the rare parameters \begin{document}$ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $\end{document} in their weight enumerators.

In the present paper, we introduce and study Goldie ADS modules and rings, which subsume two generalizations of Goldie extending modules due to Akalan et al. [3] and ADS-modules due to Alahmadi et al. [7]. A module M will be called a Goldie ADS module if for every decomposition M = S ⊕ T of M and every complement T 0 of S, there exists a submodule D of M such that T 0βD and M = S ⊕ D. Various properties concerning direct sums of Goldie ADS modules are established.

In this work, we present a number of generator matrices of the form $[I_{2n} \ | \ \tau_k(v)],$ where $I_{kn}$ is the $kn \times kn$ identity matrix, $v$ is an element in the group matrix ring $M_2(R)G$ and where $R$ is a finite commutative Frobenius ring and $G$ is a finite group of order 18. We employ these generator matrices and search for binary $[72,36,12]$ self-dual codes directly over the finite field $\mathbb{F}_2.$ As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:2102.00475; text overlap with arXiv:2102.00474

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring $$M_k(R)$$ M k ( R ) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring $$M_k(R)$$ M k ( R ) are one sided ideals in the group matrix ring $$M_k(R)G$$ M k ( R ) G and the corresponding codes over the ring R are $$G^k$$ G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes.

In this paper, a new search technique based on the virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this is a known in the literature approach due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $k^{th}$-range neighbours and search for binary $[72,36,12]$ self-dual codes. In particular, we present six generator matrices of the form $[I_{36} \ | \ \tau_6(v)],$ where $I_{36}$ is the $36 \times 36$ identity matrix, $v$ is an element in the group matrix ring $M_6(\mathbb{F}_2)G$ and $G$ is a finite group of order 6, which we then employ to the proposed algorithm and search for binary $[72,36,12]$ self-dual codes directly over the finite field $\mathbb{F}_2$. We construct 1471 new Type I binary $[72, 36, 12]$ self-dual codes with the rare parameters $\gamma=11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32$ in their weight enumerators., Comment: arXiv admin note: text overlap with arXiv:2103.07739, arXiv:2102.12863

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2] and were extensively investigated by Ahn and Anderson [1], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.

In [9], the author extends the definition of lifting and supplemented modules to ?-lifting and ?-supplemented by replacing "small submodule" with "?-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of ?-lifting and ?-supplemented modules. Especially, we show that any finite direct sum of ?-hollow modules is ?-supplemented. On the other hand, the notion of amply ?-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply ?-supplemented and satisfies Descending Chain Condition (DCC) on ?-supplemented modules and on ?-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a ?-semiperfect ring which satisfies DCC on ?-small right ideals of R.

A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a, b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study "uniquely generated modules" as a module theoretic version of "uniquely generated ideals," and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x; sigma]/(x(n+1)), where R is a strongly regular ring with an endomorphism sigma be an endomorphism of R.

For a commutative noetherian ring R , we investigate relations between tilting and cotilting modules in Mod – R and Mod – R m , where m runs over the maximal spectrum of R . For each n ω , we construct a 1 – 1 correspondence between (equivalence classes of) n -cotilting R -modules C and (equivalence classes of) compatible families F of n -cotilting R m -modules ( m ∈ mSpec ( R ) ). It is induced by the assignment C ↦ ( C m | m ∈ mSpec ( R ) ) , where C m = Hom R ( R m , C ) is the colocalization of C at m , and its inverse F ↦ ∏ F ∈ F F . We construct a similar correspondence for n -tilting modules using compatible families of localizations; however, there is no explicit formula for the inverse.

Injective, pure-injective and fp-injective modules are well known to provide for approximations in the category Mod-R for an arbitrary ring R. We prove that this fails for many other generalizations of injectivity: the $C_1$, $C_2$, $C_3$, quasi-continuous, continuous, and quasi-injective modules. We show that, except for the class of all $C_1$-modules, each of the latter classes provides for approximations only when it coincides with the injectives (for quasi-injective modules, this forces R to be a right noetherian V-ring, in the other cases, R even has to be semisimple artinian). The class of all $C_1$-modules over a right noetherian ring R is (pre)enveloping, iff R is a certain right artinian ring of Loewy length at most 2; in this case, however, R may have an arbitrary representation type.

We provide new characterizations of pseudo-Frobenius and quasi-Frobenius rings in terms of tight modules. In the process, we also provide fresh perspectives on FGF and CF conjectures. In particular, we propose new natural extensions of these conjectures which connect them with the classical theory of PF rings. Our techniques are mainly based on set-theoretic counting arguments initiated by Osofsky. Several corollaries and examples to illustrate their applications are given., To appear in Israel Journal of Mathematics

We introduce and study CLESS-modules, which subsume two generalizations of extending modules due to P.F. Smith and A. Tercan. A module $M$ will be called a CLESS-module if every closed submodule $N$ of $M$ (in the sense that $M/N$ is non-singular) with essential socle is a direct summand of $M$. Various properties concerning direct sums of CLESS-modules are established. We show that, over a Dedekind domain, a module is CLESS if and only if its torsion submodule is a direct summand. We also study the behaviour of CLESS-modules under excellent extensions of rings.

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a [Formula: see text]-group.