Nil rings and prime rings : doctoral thesis

Stopar, Nik (author), Omladič, Matjaž (mentor), Moravec, Primož (comentor)

The problem of characterizing zero product preserving maps has been studied by several authors in many different settings. Recently such maps have been considered on prime rings with nontrivial idempotents. Most of the known results assume that the map in question is bijective. In the thesis we extend these results by considering non-injective maps. More precisely, we characterize surjective additive zero product preserving maps ▫$\theta : A \to B$▫, where ▫$A$▫ is a ring with a nontrivial idempotent and ▫$B$▫ is a prime ring. We also investigate maps on rings with involution that preserve zeros of ▫$xy^\ast$▫. In particular, we obtain a characterization of surjective additive maps ▫$\theta : A \to B$▫ such that for all ▫$x,y \in A$▫ we have ▫$\theta(x) \theta(y)^\ast = 0$▫ if and only if ▫$xy^\ast = 0$▫. Here ▫$A$▫ is a unital prime ring with involution that contains a nontrivial idempotent and ▫$B$▫ is a prime ring with involution. In the second part of the thesis we devote our attention to nil rings. One of the most important open problems concerning nilrings is the Köthe conjecture, which states that a ring with no nonzero nilideals should have no nonzero nil one-sided ideals. There are many known statements that are equivalent to the Köthe conjecture and we add one more to the list. It has been proved that, when considering the validity of these statements, we may restrict ourselves to algebras over fields. We observe in the thesis that we may additionally restrict ourselves to finitely generated prime algebras. Furthermore, we investigate the connections between nilpotent, algebraic, and quasi-regular elements. It is well known that an algebraic Jacobson radical algebra over a field is nil. We generalize this result to algebras over certain PIDs and in particular to rings. On the way to this result we introduce the notion of a $\pi$-algebraic element, i.e. an element that is a zero of a polynomial with the sum of coefficients equal to one. As a corollary we show that if every element of a ring ▫$R$▫ is ▫$\pi$▫-algebraic then ▫$R$▫ is a nil ring, and at the same time obtain a new characterization of the upper nilradical. At the end we investigate the structure of the set of all ▫$\pi$▫-algebraic elements of a ring.

• Language: English
• Keywords: prime rings, zero product preservers, rings of quotients, extended centroid, involution, nil ring, upper nilradical, Jacobson radical, integral ring, ▫$\pi$▫-algebraic element, quasi-regular element, Köthe conjecture
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Place of publishing: Ljubljana
• Publisher: [N. Stopar]
• Year: 2013
• Number of pages: 94 str.
• PID: 20.500.12556/RUL-95848
• UDC: 512.552.3(043.3)
• COBISS.SI-ID: 16739161
• Publication date in RUL: 22.09.2017

Secondary language

• Language: Slovenian
• Title: Nilkolobarji in prakolobarji

Abstract

Problem karakterizacije preslikav, ki ohranjajo ničelni produkt, so študirali številni avtorji v mnogih različnih kontekstih. Nedavno so bile take preslikave obravnavane na prakolobarjih z netrivialnimi idenpotenti. Večina znanih rezultatov predpostavlja, da so omenjene preslikave bijektivne. V disertaciji razširimo te rezultate tako, da obravnavamo neinjektivne preslikave. Natančneje, podamo karakterizacijo surjektivnih aditivnih preslikav ▫$\theta : A \to B$▫, ki ohranjajo ničelni produkt, kjer je ▫$A$▫ kolobar z netrivialnim idempotentom, ▫$B$▫ pa prakolobar. Raziščemo tudi preslikave na kolobarjih z involucijo, ki ohranjajo ničle ▫$xy^\ast$▫. V posebnem karakteriziramo surjektivne aditivne preslikave ▫$\theta : A \to B$▫, za katere za vse ▫$x,y \in A$▫ velja ▫$\theta(x) \theta(y)^\ast = 0$▫ natanko tedaj, ko je ▫$xy^\ast = 0$▫. Pri tem je ▫$A$▫ enotski prakolobar z involucijo, ki vsebuje netrivialen idempotent, ▫$B$▫ pa prakolobar z involucijo. V drugem delu disertacije se posvetimo nilkolobarjem. Eden najpomembnejših odprtih problemovs področja nilkolobarjev je Köthejeva domneva, ki pravi, da kolobar brez neničelnih nilidealov nima niti neničelnih nil enostranskih idealov. Znanih je mnogo trditev, ki so ekvivalentne Köthejevi domnevi, in mi dodamo še eno na ta seznam. Dokazano je bilo, da se je za obravnavo veljavnosti teh trditev dovolj omejiti na algebre nad komutativnimi obsegi. V disertaciji opazimo, da se lahko še dodatno omejimo na končno generirane praalgebre. Poleg tega raziščemo povezave med nilpotentnimi, algebraičnimi in kvaziregularnimi elementi. Znano je, da je vsaka algebraična Jacobsonovo radikalna algebra nad komutativnim obsegom nilalgebra. Ta rezultat posplošimo na algebre nad določenimi glavnimi kolobarji in v posebnem na kolobarje. Na poti do tega rezultata vpeljemo pojem ▫$\pi$▫-algebraičnega elementa, tj. elementa, ki je ničla polinoma z vsoto koeficientov ena. Posledično dokažemo, da je kolobar, v katerem je vsak element ▫$\pi$▫-algebraičen, avtomatično nilkolobar, hkrati pa dobimo tudi novo karakterizacijo zgornjega nilradikala. Na koncu raziščemo strukturo množice vseh ▫$\pi$▫-algebraičnih elementov kolobarja.

• Keywords: prakolobar, ohranjevalci ničelnega produkta, kolobarji kvocientov, razširjeni centroid, involucija, nilkolobar, zgornji nilradikal, Jacobsonov radikal, celosten kolobar, ▫$\pi$▫-algebraičen element, kvaziregularen element, Köthejeva domneva