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An embedding theorem for free associative algebras. (English) Zbl 0729.16010

One of the many manifestations of the striking difference between commutative and noncommutative settings is the possibility of putting “big” objects inside small ones. For example, it is well-known (to the experts) that an algebra F of noncommutative polynomials (a free associative algebra) with a countable number of generators can be imbedded in the algebra G of noncommutative polynomials with two generators. The author gives a construction of such an imbedding of F into G that may be extended to an imbedding of analogs of fields of rational functions. This construction also provides a (relatively) simple example of a so-called 1-inert imbedding, which means that any factorization of \(f\in F\), considered as an element of G, is essentially a factorization in f itself.

MSC:

16K40 Infinite-dimensional and general division rings
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16S36 Ordinary and skew polynomial rings and semigroup rings
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