20.500.12556/RUL-183071 Verjetnost ničelnega množenja Probability of Zero Multiplication Verjetnostna vprašanja so že dolgo tema zanimanja v teoriji končnih grup. V tem delu bomo obravnali verjetnost, da se dva naključno izbrana elementa iz kolobarja zmnožita v 0. V prvem poglavju si bomo ogledali osnovne lastnosti te verjetnosti, anihilatorjev ter deliteljev niča. V naslednjem poglavju se bomo posvetili lokalnim kolobarjem in njihovi verjetnosti ničelnega množenja, nato pa še posebni strukturi, ki ustvari lokalen kolobar — idealizaciji. V tretjem in četrtem poglavju bomo poiskali najbolj natančne spodnje in zgornje meje za splošne kolobarje ali kolobarje z določenimi lastnostmi. Poiskali bomo tudi, kdaj so te meje stroge. Nazadnje bomo do izomorfizma natančno določili vseh 12 netrivialnih kolobarjev z verjetnostjo ničelnega množenja večjo ali enako 3/8. Večina diplomskega dela je bila narejena po zgledu članka "The probability that the multiplication of two ring elements is zero" avtorjev M. A. Esmkhani in S. M. Jafarian Amiri. Probabilistic questions have long been a topic of interest in finite group theory. In this work, we discuss the probability that two randomly chosen elements of a finite, commutative ring multiply to 0. In the first chapter, we will establish the basic properties of this probability, as well as the properties of annihilators and zero divisors. The next chapter will be dedicated to local rings and the probability of zero multiplication therein, with a subchapter about a special construction of local ring — idealization. In chapters three and four, we will find the best upper and lower bounds for the probability of zero multiplication in finite, commutative rings in general as well as in some rings with particular properties. We will also prove for which rings these bounds are sharp. Finally, we will determine all 12 non-trivial rings up to an isomorphism, whose probability of zero multiplication is greater than or equal to 3/8. The majority of this thesis was based upon the findings in the article "The probability that the multiplication of two ring elements is zero" by the authors M. A. Esmkhani and S. M. Jafarian Amiri. komutativna algebra delitelji niča končni kolobarji commutative algebra zero divisors finite rings true false false Slovenski jezik Angleški jezik Diplomsko delo/naloga 2026-06-03 12:05:05 2026-06-03 12:05:14 2026-06-10 06:48:34 0000-00-00 00:00:00 2026 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 38752 281048323 Perovsek_Aljaz_-_Verjetnost_nicelnega_mnozenja.pdf Perovsek_Aljaz_-_Verjetnost_nicelnega_mnozenja.pdf 1 1592445328EE7628619DC58928D3BEFB b3a02fbb6b8bcccf060f932578473a024bb236d7d5e652e42ee7ffbac6f82d97 593e5f64-5f33-11f1-b0ab-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=234901 Fakulteta za računalništvo in informatiko 0 0 0