20.500.12556/RUL-174022 Nil-čisti kolobarji magistrsko delo Nil-clean rings Delo obravnava nil-čiste kolobarje, v katerih je vsak element vsota idempotenta in nilpotenta. Posebej so izpostavljeni strogo nil-čisti kolobarji, za katere je kvocient po Jacobsonovem radikalu Boolov, radikal pa nilideal, in enolično nil-čisti kolobarji, kjer ima vsak element natanko en nil-čisti razcep. Posvetimo se tudi matričnim kolobarjem in dokažemo, da je $M_n(F)$ nil-čist natanko tedaj, ko je polje $F \cong \mathbb{F}_2$. V zadnjem poglavju analiziramo verjetnost, da je naključno izbran element končnega komutativnega kolobarja nil-čist, in pokažemo, da je v kolobarju $\mathbb{Z}_{p^k}$ delež nil-čistih elementov enak $\frac{2}{p}$. S tem ugotovimo, da so kolobarji $\mathbb{Z}_{2^k}$ vedno nil-čisti, pri lihih praštevilih $p$ pa ta lastnost odpade. Rezultati kažejo, da je teorija nil-čistih kolobarjev tesno povezana z Boolovimi strukturami in radikali. This thesis studies nil-clean rings, in which every element can be expressed as the sum of an idempotent and a nilpotent element. Special attention is given to strongly nil-clean rings, characterized by the property that the quotient modulo the Jacobson radical is Boolean and the radical itself is a nilideal, as well as to uniquely nil-clean rings, where every element has exactly one nil-clean decomposition. We also examine matrix rings and prove that $M_n(F)$ is nil-clean if and only if the field $F$ satisfies $F \cong \mathbb{F}_2$. In the final part, we analyze the probability that a randomly chosen element of a finite commutative ring is nil-clean, and show that in $\mathbb{Z}_{p^k}$ the proportion of nil-clean elements equals $\frac{2}{p}$. It follows that the rings $\mathbb{Z}_{2^k}$ are always nil-clean, while for odd primes $p$ this property does not hold. The results demonstrate that the theory of nil-clean rings is closely related to Boolean structures and radicals. Nil-čisti kolobarji čisti kolobarji idempotenti nilpotenti Jacobsonov radikal matrični kolobarji komutativni kolobarji Nil-clean rings clean rings idempotents nilpotents Jacobson radical matrix rings commutative rings true false false Slovenski jezik Angleški jezik Magistrsko delo/naloga 2025-09-26 08:15:05 2025-09-26 08:15:09 2025-11-16 04:01:06 0000-00-00 00:00:00 2025 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512 154761 250565379 19025.pdf 19025.pdf 1 BADAB4570A33773B70C48B625FDABA36 17aea5b5302a91fd5eb0ada500e3bb6b1ed477ba9ed55463f445c33243c08baa f6a12abf-9a9f-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=218760 Fakulteta za matematiko in fiziko 0 0 0