Graf deliteljev niča komutativnega kolobarja

Petauer, Eva

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Title:	Graf deliteljev niča komutativnega kolobarja
Authors:	ID Petauer, Eva (Author) ID Grašič, Mateja (Mentor) More about this mentor...
Files:	MAG_Petauer_Eva_2025.pdf (1,16 MB) MD5: 08342885377894C556D5CE3E10A44000
Language:	Slovenian
Work type:	Master's thesis/paper
Typology:	2.09 - Master's Thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Graf deliteljev niča $\Gamma(R)$ komutativnega kolobarja $R$ povezuje teorijo kolobarjev s teorijo grafov. Vozlišča grafa deliteljev niča so neničelni delitelji niča, povezava med vozliščema $x$ in $y$ pa obstaja, kadar je $xy=0$. Cilj naloge je raziskati, kako lastnosti kolobarja (noetherskost, artinskost, lokalnost, končnost) vplivajo na strukturo grafa deliteljev niča ter obratno, katere informacije o kolobarju lahko razberemo iz grafa deliteljev niča. Vsak graf na vsaj treh vozliščih lahko predstavimo kot graf deliteljev niča nekega kolobarja, za grafe na več kot treh vozliščih pa to ne drži. Velja, da je vsak graf deliteljev niča nekega komutativnega kolobarja povezan in da ima premer manjši od štiri. Če vsebuje cikel, je dolžina najkrajšega cikla manjša ali enaka sedem. Če obravnavamo komutativne artinske kolobarje, katerih graf deliteljev niča vsebuje cikel, je dolžina najkrajšega cikla manjša ali enaka štiri. V grafu deliteljev niča obstaja vozlišče, ki je povezano z vsakim drugim vozliščem natanko takrat, ko je bodisi $R \cong \mathbb{Z}_2 \times A$, pri čemer je $A$ cel kolobar bodisi je $Z_0(R)$ anihilatorski ideal. Obravnavamo še polne grafe in pokažemo, da če je $\Gamma(R)$ polni graf, potem je bodisi $R \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ bodisi je $R$ lokalni kolobar. Na koncu še dokažemo, da je graf deliteljev niča z več kot tremi vozlišči graf zvezda natanko takrat, ko je $R \cong \mathbb{Z}_2 \times F$, pri čemer je $F$ končno polje. Velja tudi, da ima graf zvezda $p^n$ vozlišč, pri čemer je $p$ neko praštevilo in $n \in \mathbb{N}_0$ in obratno, vsak graf zvezda velikosti $p^n$ je lahko graf deliteljev niča.
Keywords:	komutativen kolobar, delitelj niča, graf deliteljev niča
Place of publishing:	Maribor
Publisher:	[E. Petauer]
Year of publishing:	2025
PID:	20.500.12556/DKUM-95159
UDC:	519.17(043.2)
COBISS.SI-ID:	252338435
Publication date in DKUM:	08.10.2025
Views:	198
Downloads:	50
Metadata:
Categories:	FNM
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Licences

License:	CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International

Link:	http://creativecommons.org/licenses/by-nc-nd/4.0/
Description:	The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.
Licensing start date:	08.09.2025

Secondary language

Language:	English
Title:	The Zero-Divisors Graph of a Commutative Ring
Abstract:	The zero-divisor graph $\Gamma(R)$ of a commutative ring $R$ connects ring theory with graph theory. The vertices of the zero-divisor graph are the nonzero zero-divisors, and two vertices $x$ and $y$ are adjacent whenever $xy=0$. The aim of this work is to analyze how properties of the ring (Noetherian, Artinian, local, finite) influence the structure of the zero-divisor graph, and conversely, what information about the ring can be deduced from the zero-divisor graph. Every graph with at least three vertices can be represented as the zero-divisor graph of some ring, but this does not hold for graphs with more than three vertices. It is known that every zero-divisor graph of a commutative ring is connected and has diameter less than four. If it contains a cycle, then the length of the shortest cycle is at most seven. In the case of commutative Artinian rings whose zero-divisor graph contains a cycle, the length of the shortest cycle is at most four. A vertex in the zero-divisor graph is adjacent to every other vertex if and only if either $R \cong \mathbb{Z}_2 \times A$ with $A$ an integral domain, or $Z_0(R)$ is an annihilator ideal. We also consider complete graphs and show that if $\Gamma(R)$ is complete, then either $R \cong \mathbb{Z}_2 \times \mathbb{Z}_2$ or $R$ is a local ring. Finally, we prove that the zero-divisor graph with more than three vertices is a star graph if and only if $R \cong \mathbb{Z}_2 \times F$, where $F$ is a finite field. It also holds that a star graph has $p^n$ vertices, where $p$ is a prime and $n \in \mathbb{N}_0$, and conversely, every star graph of size $p^n$ can occur as a zero-divisor graph.
Keywords:	commutative ring, zero divisor, zero divisor graph

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