On graded radical ideals of graded non-commutative rings (Q6638241)

The study of \textit{graded rings} and their associated \textit{radicals} has become one of the fundamental areas of research in ring theory. A \(G\)-graded ring \(R\) is defined by its decomposition into additive subgroups indexed by elements of a group \(G,\) with multiplication satisfying:\N\[\NRgRh \subseteq R_{gh} \quad \text{for all } g, h \in G.\N\]\NWhen \(R\) is commutative, previous works have introduced various ideals, such as graded \(n\)-ideals and graded \(J\)-ideals, corresponding to the graded prime and Jacobson radicals, respectively. These concepts were extensively studied for their algebraic and structural properties. In this paper, the authors extend these concepts to non-commutative graded rings by introducing a general framework of graded \(\rho_G\)-ideals, where \(\rho_G\) is a graded special radical. This generalization provides a unified theory encompassing the existing results for commutative rings as special cases. The article presents the following results:\N\N\begin{itemize}\N\item[1.] Graded \(\rho_G\)-ideals and their properties. A \textit{graded \(\rho_G\)-ideal} is defined as a graded ideal \(I\) of a \(G\)-graded ring \(R\) such that for homogeneous elements \(a, b \in R\):\N\[\NaRb \subseteq I \quad \text{and} \quad a \notin \rho_G(R) \text{ implies } b \in I.\N\]\NThe authors prove that graded \(n\)-ideals (connected to the prime radical) and graded \(J\)-ideals (connected to the Jacobson radical) are specific instances of graded \(\rho_G\)-ideals. The paper establishes that for any graded special radical \(\rho_G\), the intersection of all graded \(\rho_G\)-ideals of \(R\) coincides with \(\rho_G(R)\).\N\N\item[2.] Maximal graded \(\rho_G\)-ideals. A maximal graded \(\rho_G\)-ideal is introduced as a proper graded \(\rho_G\)-ideal that is not properly contained in any other graded \(\rho_G\)-ideal. The authors show that every maximal graded \(\rho_G\)-ideal is a graded prime ideal. If \(\rho_G(R)\) is itself a graded prime ideal, it is also maximal.\N\item[3.] Graded \(\rho_G\)-m-systems The concept of a \textit{graded \(\rho_G\)-m-system} is introduced. A subset \(S\) of \(R\) is a graded \(\rho_G\)-m-system if:\N\[\NtRs \cap S \neq \emptyset \quad \text{for all } t \in h(R) - \rho_G(R) \text{ and } s \in S.\N\]\NIt is proven that a proper graded ideal \(I\) of \(R\) is a graded \(\rho_G\)-ideal if and only if the complement \(h(R) - I\) forms a graded \(\rho_G\)-m-system.\N\item[4.] Graded \(\mathcal{J}_G\)-ideal When \(\rho_G\) is the \textit{graded prime radical}, the authors recover results on graded \(n\)-ideals, showing that the zero ideal in a graded prime ring is a graded \(n\)-ideal. For the \textit{graded Jacobson radical}, the paper characterizes graded \(J\)-ideals in terms of their correspondence with maximal graded ideals and proves that they are equivalent to graded \(J\)-primary ideals when contained in the Jacobson radical.\N\end{itemize}