Анализа компутативних прстена придруживањем симплицијалних комплекса
Analysis of commutative rings by associating simplical complexes.
Doktorand
Milošević, NelaMentor
Petrović, ZoranČlanovi komisije
Lipkovski, AleksandarPetrić, Zoran
Pucanović, Zoran
Metapodaci
Prikaz svih podataka o disertacijiSažetak
Predmet izuqavaa doktorske disertacije su simplicijalni kompleksi
pridrueni komutativnim prstenima sa jedinicom. Generalno, kombi-
natorni objekti mogu biti pridrueni prstenima na razliqite naqine,
i u ovoj disertaciji izuqavamo vixe simplicijalnih kompleksa koji
daju interesantne rezultate. Fokus rada je odreivae homotopskog
tipa geometrijske realizacije takvih simplicijalnih kompleksa u slu-
qajevima kada je to mogue.
Za djelimiqno ureen skup netrivijalnih ideala u komutativnom
prstenu, definixe se ureajni kompleks i odreuje egov homotopski
tip u generalnom sluqaju.
Simplicijalni kompleks moe biti i indirektno pridruen prstenu,
kao kompleks nezavisnosti nekog grafa ili hipergrafa koji je pridru-
en prstenu. Za komaksimalan graf definixemo egov kompleks neza-
visnosti i odreujemo homotopski tip za generalne komutativne prstene
sa jedinicom.
Da e, ova teza se bavi i izuqavaem nula djelite a tako xto se po-
smatraju ideali koji su nula djelite i i definixe se kompleks ideala
nula djelit...e a. Homotopski tip ovog simplicijalnog kompleksa odre-
uje se za konaqne prstene kao i za prstene sa beskonaqno mnogo mak-
simalnih ideala. U ovom dijelu koristi se diskretna teorija Morsa
za simplicijalne komplekse. Teoreme dokazane u disertaciji primje-
ujemo na neke klase komutativnih prstena qime dolazimo do intere-
santnih kombinatornih rezultata.
This dissertation examines simplicial complexes associated with commutative
rings with unity. In general, a combinatorial object can be attached to a ring in
many dierent ways, and in this dissertation we examine several simplicial complexes
attached to rings which give interesting results. Focus of this thesis is determining
the homotopy type of geometric realization of these simplicial complexes, when it is
possible.
For a partially ordered set of nontrivial ideals in a commutative ring with identity,
we investigate order complex and determine its homotopy type for the general case.
Simplicial complex can also be associated to a ring indirectly, as an independence
complex of some graph or hypergraph which is associated to that ring. For
the comaximal graph of commutative ring with identity we dene its independence
complex and determine its homotopy type for general commutative rings with identity.
This thesis also focuses on the study of zero-divisors, by investigating ideals
which a...re zero-divisors and dening zero-divisor ideal complex. The homotopy type
of geometric realization of this simplicial complex is determined for rings that are
nite and for rings that have innitely many maximal ideals. In this part of the
thesis, we use the discrete Morse theory for simplicial complexes. The theorems
proven in this dissertation are then applied to certain classes of commutative rings,
which gives us some interesting combinatorial results.