Generating ideals by additive subgroups of rings
نویسندگان
چکیده
We obtain several fundamental results on finite index ideals and additive subgroups of rings as well model-theoretic connected components rings, which concern generating in finitely many steps inside groups rings. Let R be any ring equipped with an arbitrary additional first order structure, A a set parameters. show that whenever H is A-definable, subgroup (R,+), then H+R⋅H contains two-sided ideal index. As corollary, we positive answer to Question 4.9 [4]: if unital, (R¯,+)A00+R¯⋅(R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00, also implies R¯A00=R¯A000, where R¯≻R sufficiently saturated elementary extension R, (R¯,+)A00 [resp. R¯A00] denotes the smallest A-type-definable, bounded ideal] R¯, R¯A000 invariant over A, R¯. If characteristic (not necessarily unital), get sharper result: (R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00. similar result (but more required) for generated unital) analogous topological The above unital simplified descriptions definable (and so classical) Bohr compactifications triangular obtained Corollary 4.5 [4] are valid all analyze concrete examples, compute number needed generate group by (R¯∪{1})⋅(R¯,+)A00 study related aspects, showing “optimality” some our main yielding answers natural questions.
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ژورنال
عنوان ژورنال: Annals of Pure and Applied Logic
سال: 2022
ISSN: ['0168-0072', '1873-2461']
DOI: https://doi.org/10.1016/j.apal.2022.103119