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.
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.
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.
<p style='text-indent:20px;'>In this paper, we show that one can construct a <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes are reversible of index <inline-formula><tex-math id="M5">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>. Additionally, we introduce a new family of rings, <inline-formula><tex-math id="M6">\begin{document}$ {\mathcal{F}}_{j,k} $\end{document}</tex-math></inline-formula>, whose base is the finite field of order <inline-formula><tex-math id="M7">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> and study reversible <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes over this family of rings. Moreover, we present some possible applications of reversible <inline-formula><tex-math id="M9">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes over <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{F}}_{j,k} $\end{document}</tex-math></inline-formula> to reversible DNA codes. We construct many reversible <inline-formula><tex-math id="M11">\begin{document}$ G $\end{document}</tex-math></inline-formula>-codes over <inline-formula><tex-math id="M12">\begin{document}$ {\mathbb{F}}_4 $\end{document}</tex-math></inline-formula> of which some are optimal. These codes can be used to obtain reversible DNA codes.</p>
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