Permutation identities and fractal structure of rings (Q6547694)

The authors introduce the notion of a product fractal ideal of a ring as a generalization of a ring ideal. This is done by using permutations of finite sets and the multiplication in the ring. For a ring \(R\) and a natural number \(k,\) let \(R_{k}\) be the subring of \(R\) defined by\N\[\NR_{k}=\left\{ \sum_{r=1}^{m}(a_{r1}a_{r2}\cdots a_{rk})\mid m\in\mathbb{N},a_{ri}\in R,1\leq r\leq m,1\leq i\leq k\right\} .\N\]\NFor a permutation \(\rho \) from the permutation group \(S_{k}\) and \(A\) a nonempty subset of \(R,\) let \(R_{k,\rho ,A}\) be the subring of \(R\) generated by the set \(\{a\in R_{k}\mid a\) is \(\rho \)-representable with respect to \(A\}.\) Here \(a\in R_{k}\) is called \(\rho \)-representable with respect to \(A\) if there is an \(m\in\mathbb{N}\) and \(a_{ri}\in R,1\leq r\leq m,1\leq i\leq k\) such that \(a=\sum_{r=1}^{m}(a_{r1}a_{r2}\dots a_{rk})\) with \(-\sum_{r=1}^{m}(a_{r1}a_{r2}\dots a_{rk})+\sum_{r=1}^{m}(a_{r\rho (1)}a_{r\rho (2)}\dots a_{r\rho (k)})\in A.\) A subgroup \(J\) of \((R,+)\) is called a \((k,\rho ,A)\)\textit{-product fractal ideal of }\(R\) if \(R_{k,\rho,A}J\subseteq J\) and \(JR_{k,\rho ,A}\subseteq J\).\N\NSeveral examples are given; mostly for matrix rings. A product fractal ideal determines an equivalence relation on a substructure of the ring. This gives rise to a corresponding quotient ring, which partitions the ring under certain conditions. Product fractal homomorphisms are defined and fractal isomorphism theorems for rings are defined which extend the classical isomorphism theorems. The structure given by the fractal isomorphism theorems is infinitely repetitive and shows a fractal pattern.\NAll the quotients are formed by the kernel of a single product fractal homomorphism.