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Ascending chain conditions on principal left and right ideals for semidirect products of semigroups
Stopar, Nik
V članku raziščemo pogoje naraščajočih verig glavnih levih in desnih idealov za poldirektne produkte polgrup in pokažemo, kako je to povezano s pripadajočim problemom za kolobarje poševnih ...posplošenih potenčnih vrst. Naj bo ▫$S$▫ levo krajšljiva polgrupa z enoličnim idempotentom ▫$e, T$▫ desno krajšljiva polgrupa z idempotentom ▫$f$▫ in ▫$\omega: T \to \operatorname{End}(S)$▫ homomorfizem polgrup, za katerega velja ▫$\omega(f) = id_S$▫. Dokažemo, da v tem primeru poldirektni produkt ▫$S \rtimes_\omega T$▫ zadošča pogoju naraščajočih verig glavnih levih idealov (oz. desnih idealov) natanko tedaj, ko ▫$S$▫ in ▫$T$▫ zadoščata pogoju naraščajočih verig glavnih levih idealov (oz. desnih idealov in je ▫$\operatorname{Im} \omega(t)$▫ zaprta za popolne inverze za vse ▫$t \in T$▫). Podamo tudi več primerov, ki pokažejo, da za bolj splošne polgrupe te implikacije ne veljajo nujno.
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