Nil rings and prime rings - Stopar, Nik (visokošolska dela, 2013)
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 ?$θ : A → B$?, 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 ?$xy^*$?. In particular, we obtain a characterization of surjective additive maps ?$θ : A → B$? such that for all ?$x,y \in A$? we have ?$θ(x) θ(y)^* = 0$? if and only if ?$xy^* = 0$?. 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 $π$-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 ?$π$?-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 ?$π$?-algebraic elements of a ring.
Keywords: algebraic elements; algebraični elementi; celosten kolobar; Disertacije; extended centroid; integral ring; involucija; involution; Jacobson radical; Jacobsonov radikal; Kolobarji (matematika); kolobarji kvocientov; Köthe conjecture; Köthejeva domneva; kvaziregularen element; nil ring; nilkolobar; ohranjevalci ničelnega produkta; prakolobar; prime rings; quasi-regular element; razširjeni centroid; rings of quotients; upper nilradical; zero product preservers; zgornji nilradikal