Rational operators of the space of formal series

Some results relevant to embeddability of rings (especially group algebras) in division rings

“…Thus cl R n (S) = R n , so since (15) implies (iii), there exists S 0 ⊆ S of cardinality < n such that cl R n (S 0 ) = cl R n (S). Hence u / ∈ cl R n (S 0 ) and u ∈ cl R n (S 0 ∪ {t}); so (15) gives t ∈ cl R n (S 0 ∪ {u}), whence t ∈ cl R n (S ∪ {u}), establishing (12).…”

“…In the situations we shall be looking at, conditions (9)-(11) will generally be easy to establish. The next lemma restricts the instances one has to verify to show that (12) also holds, and shows that (13) is implied by (9) and (12). Lemma 2.…”

“…(11) For every homomorphism of right R-modules h : R m → R n and every cl R n -closed submodule A ⊆ R n , the submodule h −1 (A) ⊆ R m is cl R m -closed. (12) The closure operator cl R n has the exchange property, namely, for S ⊆ R n and t, u ∈ R n , if u / ∈ cl R n (S) but u ∈ cl R n (S ∪ {t}), then t ∈ cl R n (S ∪ {u}).…”

On embedding left-ordered groups into division rings

“…To improve the readability of this paper and make it self-contained, we will carefully quote and describe any definition and theorem from [12][13][14] which are needed to understand the general construction. All what follows in this section is taken from [12][13][14][15][16]18] and can be found there at different places, most of them exactly the way we will use it here or a little changed and hidden. Obviously, the sum and the difference of two continuous endomorphisms is also continuous, but it is not easy to prove that the continuous and v-compatible automorphisms of F ..// form a subgroup of Aut F ..//.…”

“…This idea of embedding groups into skew fields has also been generalized by Dubrovin to groups with cones which need not be left-orderable (cf. [13,16]). In any version of his general embedding theorem G acts on an F -vector space F ..// of all formal power series over F where F is a skew field and a linearly ordered set.…”