A factorization theory for some free fields.

*(English)*Zbl 07251171Let \(\mathbb{K}\) 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.

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.

Reviewer: Wen-Fong Ke (Tainan)

##### 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.) |

##### Keywords:

free associative algebra; factorization of non-commutative polynomials; minimal linear representation; universal field of fractions; admissible linear system; non-commutative formal power series
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