Exponential rings, exponential polynomials and exponential functions (Q1082388)

An exponential ring is a pair (R,E), where R is a commutative ring with unit 1 and \(E: (R,+)\to U\) (the multiplicative group of units in R) satisfying \(E(x+y)=E(x)E(y)(x,y\in R),E(0)=1.\) In preparation for proving that each subset of \({\mathbb{R}}^ 2\) definable (with parameters) in the language of exponential rings has finitely many connected components, the author deals with the following questions. What is the structure of the free exponential rings and of the free algebras over a given exponential ring ? When is a map injective which assigns to a polynomial in \(x_ 1,...,x_ m\) over (R,E) the corresponding R-valued m-place function ?