Complexity of elements in rings (Q1868341)

For an associative ring \(R\) with identity element the symbol \(U(R)\) denotes the group of units of \(R\). If \(S\) is a subset of \(R\), then the rational closure \(D\) of \(S\) in \(R\) is the smallest subring of \(R\) containing \(S\) and such that \(U(D)=D\cap U(R)\). In the second part the authors present a new approach to the complexity of elements in \(D\). This allows to give new proofs of all important results concerning the structure of \(D\). The third part investigates the interaction between complexity and addition, while the last one is devoted to complexity and multiplication.