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A factorization theory for some free fields. (English) Zbl 07251171
Let $$\mathbb{K}$$ be a commutative field and $$X=\{x_{1},\dots,x_{d}\}$$ a non-empty finite alphabet. Then the free associative algebra $$R=\mathbb{K}\langle X\rangle$$ is a similarity unique factorization domain. The elements in its universal field of fractions, $$\mathbb{F}=\mathbb{K}(\langle X\rangle)$$, can be represented by linear systems, and rational operations (scalar multiplication, addition, multiplication, inverse) can be easily formulated in terms of linear representations [P. M. Cohn and C. Reutenauer, Can. J. Math. 46, No. 3, 517–531 (1994; Zbl 0836.16012)].
With a proper definition of left (and right) divisibility based on the rank of an element, a factorization theory on $$\mathbb{F}$$ is established on the level of minimal linear representations. This extends the factorization in $$R$$. A concrete approach to factorize elements into their (generalized) atoms is presented in the last section.
##### MSC:
 16K40 Infinite-dimensional and general division rings 16Z05 Computational aspects of associative rings (general theory) 16G99 Representation theory of associative rings and algebras 16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
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