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Commutation Problems on Sets of Words and Formal Power Series
, 2002
"... We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generaliza ..."
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We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series  the semilinear and the bounded series  both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language  the largest set commuting with a language , of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago  asking whether or not the centralizer of any rational language is rational  in the case of periodic, binary, and ternary sets of words, as well as for rational ccodes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an ccode X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.
major topic: “Noncommutative Rings” 1 Fundamental Ring Theory
, 2002
"... Semisimple rings: definitions and examples; homological characterizations; WedderburnArtin theorem (includ. Rieffel’s approach); simple rings (artinian and nonartinian). ([Lam91] §§1–3) Jsemisimplicity: characterizations of the Jacobson radical; examples; artinian Jsemisimple rings; HopkinsLevit ..."
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Semisimple rings: definitions and examples; homological characterizations; WedderburnArtin theorem (includ. Rieffel’s approach); simple rings (artinian and nonartinian). ([Lam91] §§1–3) Jsemisimplicity: characterizations of the Jacobson radical; examples; artinian Jsemisimple rings; HopkinsLevitzki theorem; Nakayama’s lemma; von Neuman regular rings as nearly semisimple rings. ([Lam91] §4)