Embeddings into power series rings
Let S be a totally ordered semigroup such that \(a<b\) implies \(ac<bc\) and \(ca<cb\) for all c in S. Let r be a function defined on \(S\times S\) with values in the positive reals such that \(r(ab,c)\cdot r(a,b)=r(a,bc)\cdot r(bc)\) for all a, b, c in S. Then the power series ring \({\mathbb{R}}((S))\) is defined as follows: The elements of \({\mathbb{R}}((S))\) are the functions \(f: S\to {\mathbb{R}}\) whose support \(s(f)=\{a\in S\); f(a)\(\neq 0\}\) is inversely well-ordered in S. Addition of two such functions f and g is defined pointwise and multiplication by \[ (f\cdot g)(c)=\sum_{ab=c}r(a,b)f(a)g(b). \] \({\mathbb{R}}((S))\) with the lexicographic order \(f>0\) iff f(max s(f))\(>0\) becomes a totally ordered ring without zero divisors. It is well known that every totally ordered commutative field can be order embedded in some \({\mathbb{R}}((S))\). This paper presents a valuable contribution to the question which totally ordered rings without zero divisors are order embeddable in some \({\mathbb{R}}((S))\). An interesting consequence is the following: Every totally ordered division ring for which the group of archimedean classes is isomorphic to \(({\mathbb{Z}},+)\) is embeddable in the above sense.
scientific article; zbMATH DE number 3991568
scientific article