Endlichwertige Vielfachenketten in Ringen mit Associativungleichungen (Q1215697)

The following concept of a generalized ring is introduced: \(R\) is a generalized ring if \(R\) satisfies all the axioms of a ring, but the associative law for multiplication is replaced by the following inequalities: \[ x(yR) x(RY) (xy)R+ (xR)y \tag{A} \] \[ (xR)Y+ (Rx)Y cx(RY) + R(xy) \tag{B} \] \[ (xy)R+ (Yx)R c-x(YR) + Y(xR) \text{ for all }x,y \text{ in }R. \tag{C} \] Ordinary associative rings and Lie-rings are generalized rings. Additional examples are constructed as follows: Let \((G,+)\) be an abelian group, \(\Gamma\) a subgroup of \((\operatorname{End}G,+)\). We take \((R,+) = G\oplus\Gamma\), the direct sum of \(G\) and \(\Gamma\), and define multiplication on \(R\) by \[ (g + \sigma)\cdot (g' + \sigma') =g^{\sigma'} + g'^{\sigma},\text{ with }g, g'\text{ in }G,\ a, a'\text{ in }\Gamma. \] Then \(R\) is a generalized ring if and only if for all \(g\) in \(G\), \(\sigma', \sigma''\) in \(\Gamma\) there exists an \(h\) in \(G\), \(\sigma\) in \(\Gamma\) with \[ g^{\sigma'\sigma''} = h^{\sigma''\sigma'} = g^{\sigma\sigma'}. \] In the following we will say `ring' instead of `generalized ring'. A sequence of elements \(r_1, r_2,\ldots\) in a ring \(R\) is called a sequence of multiples if \(r_{i+1}\) is in \(r_iR\) for all \(i\). Two main problems are investigated: 1) To characterize the elements \(r\) in \(R\) with the property that every sequence of multiples beginning with \(r\) contains a finite number of distinct elements only; 2) To describe the ideals \(J\) of \(R\) which have the property that the number of distinct elements in a sequence of multiples contained in \(J\) is bounded. To formulate some of the results we need the following definitions: Let \(R\) be a ring; \(\mathfrak AR = \{x\in R;\ xR= 0\}\); \(\mathfrak{HA}R = \cap X\), taken over all ideals \(X\) of \(R\) with \(\mathfrak A(R/X) =0\); \(rR^\omega = \) subset \(S\) of \(R\) minimal with the property that \(r\) in \(S\), and \(x\) in \(S\) implies \(xt\) in \(S\) for any element \(t\) in \(R\); \(\mathfrak PR = \{r\text{ in }R;\ \vert rR^\omega\vert < \infty\}\); \(\mathfrak CR = \) ideal of \(R\) with \(\mathfrak CR\supseteq \mathfrak PR\) and \(\mathfrak CR/\mathfrak PR = \mathfrak{HA}(R/\mathfrak PR)\). We say \(r\) in \(R\) is a \(W\)-element if every sequence of multiples beginning with \(r\) contains finitely many different elements only; \(r\) is a \(V\)-element if every sequence of multiples \(\{r_i\}\) beginning with \(r\) contains at least one pair \(r_i\), \(r_j\) with \(i\ne j\) and \(r_i =r_j\). Theorem: \(\mathfrak PR = \{r\text{ in }R\); \(r\) is \(V\)-element and \(\vert rR\vert < \infty\}\). Theorem: \(\mathfrak CR = \) set of all \(W\)-elements = set of all \(V\)-elements. Now some results concerning problem 2). The following conditions are equivalent for a right ideal \(J\) of \(R\): (i) There exists a positive integer \(b\) such that every sequence of multiples contained in \(J\) contains at most \(b\) different elements. (ii) There exists a positive integer \(c\) such that for every sequence \(\{r_i\}\) of multiples with \(r_i\) in \(J\) for all \(i\) a pair of integers \(j\), \(k\) exists with \(1\le k<j\le c\) and \(r_k = r_j\). (iii) There exists a right ideal \(B\) contained in \(J\) such that for some positive integer \(h\) the \(h\)-th term of a sequence of multiples contained in \(J\) is contained in \(B\). Further, every element of \(B\) is contained in a finite right ideal of bounded order. Then elements \(r\) with the following property are characterized: There exist positive integers \(a\), \(b\) such that the right ideal generated by \(x\) is finite with order dividing \(b\), provided \(x\) is the \(a\)-th term in a sequence of multiples beginning with \(r\). We finally mention the hierarchy of EK-elements. Here, \(r\) in \(R\) is called an EK-element if there exists a positive integer \(k\) such that for every sequence \(\{r_i\}\) of multiples starting with \(r\), a pair of indices \(j\), \(t\) with \(1\le j<t\le k\) and \(r_j = r_t\) exists. The hierarchy consists of seven subsets of the set of all EK-elements each contained in the next one. We give one illustration: We say \(r\) in \(R\) is a BW-element if there exists a positive integer \(k\) such that every sequence of multiples starting with \(r\) has at most \(k\) different members. It is clear that a BW-element is an EK-element, it remains an open question if the converse holds.