# Embedding domains in division rings

Javier Sánchez Serdà1

Università degli Studi dell'Insubria,
Universitat Autònoma de Barcelona

Suppose we have a domain embedded in a division ring . We define inductively:
, and for
.
Then is the smallest division ring that contains inside . We define , the inversion height of inside as if there is no such that is a division ring. Otherwise,

Let be a commutative field. Suppose is a ring homomorphism which is not onto. Let and Consider the skew polynomial ring It was proved by Jategaonkar that the -algebra generated by and is a free -algebra We call these embeddings Jategaonkar embeddings.

We prove:

Jategaonkar embeddings have at most inversion height And there are examples of Jategaonkar embeddings of height one and two.

If there is an embedding of the free algebra on two generators of height then there exists an embedding of the free algebra on an infinite number of generators of inversion height

Let be the free algebra or the free group algebra on generators. We use examples in to obtain embeddings of of inversion height or

In a Jategaonkar embedding is never the universal field of fractions of

#### Footnotes

...à1
joint work with D. Herbera

2005-05-23