Let R be a ring with 1 and assume that the elements \(y_ 1,...,y_ n\) in R generate freely a right ideal. Each \(y_ jR\) contains the right ideal generated freely by \(y_ jy_ 1,...,y_ jy_ n\), and a similar thing applies to the ideals \(y_ jy_ iR\), and so on. This structure enables us to construct right ideals we name fractal ideals. There are two kinds of fractal ideals: decreasing and increasing. Starting with \(y_ 1R+...+y_ nR\) a decreasing fractal ideal is built by replacing some of the summands \(y_ jR\) by \(y_ jy_ 1R+...+y_ jy_ nR\), then some of the summands \(y_ jy_ iR\) are replaced by \(y_ jy_ iy_ 1R+...+y_ jy_ iy_ nR\) and so on, letting the process to be finite or infinite. In this way we get a sequence of decreasing right ideals and the fractal ideal is defined to be their intersection. This structure may be best thought of in terms of rooted n-ary directed trees, i.e., in which each node is either a leaf or has n sons. The element 1 is the root, its sons are \(y_ 1,...,y_ n\), and if yR is replaced by \(yy_ 1R+...+yy_ nR\) then the node representing y is connected to the nodes \(yy_ 1,...,yy_ n.\)
Increasing fractal ideals are the union of an increasing sequence of right ideals \(I_ k\). Looking at n-ary trees as above, \(I_ k\) is the right ideal generated by the elements represented by the leaves of the tree of depth \(\leq k.\)
In this paper we investigate the property of essentiality in fractal ideals in the group ring KG of the free group \(G=<x_ 1,...,x_ n>\). The ideal involved is the augmentation ideal and the “special” base is \(\{1-x_ 1,...,1-x_ n\}\). We examine infinite trees which have at each depth only one node which is not a leaf. Such a tree can be represented by a sequence \(d=(d_ 1,d_ 2,d_ 3,...)\), \(1\leq d_ i\leq n\), which tells us where at each step the single replacement takes place. Denoting the increasing fractal ideal corresponding to the sequence d by \(I_ d\), our main result is that \(I_ d\) is essential if and only if d is not eventually constant. As for decreasing fractal ideals, it is stated (but not proved here), that the ideal corresponding to d is always essential, regardless of d.
Our original motivation was to have interesting “nontrivial” examples of essential ideals in order to better understand the maximal ring of fractions.