Rank of elements of general rings in connection with unit-regularity

We define the rank of elements of general unital rings, discuss its properties and give several examples to support the definition. In semiprime rings we give a characterization of rank in terms of invertible elements. As an application we prove that every element in the socle of a unital semiprime ring is unit-regular.Comment: 13 page

Final solution to the problem of relating a true copula to an imprecise copula

In this paper we solve in the negative the problem proposed in this journal (I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48-66) whether an order interval defined by an imprecise copula contains a copula. Namely, if $\mathcal{C}$ is a nonempty set of copulas, then $\underline{C} = \inf\{C\}_{C\in\mathcal{C}}$ and $\overline{C}= \sup\{C\}_{C\in\mathcal{C}}$ are quasi-copulas and the pair $(\underline{C},\overline{C})$ is an imprecise copula according to the definition introduced in the cited paper, following the ideas of $p$-boxes. We show that there is an imprecise copula $(A,B)$ in this sense such that there is no copula $C$ whatsoever satisfying $A \leqslant C\leqslant B$. So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of Dibala et al. (Defects and transformations of quasi-copulas, Kybernetika, 52 (2016), 848-865) where possibly negative volumes of quasi-copulas as defects from being copulas were studied.Comment: 20 pages; added Conclusion, added some clarifications in proofs, added some explanations at the beginning of each section, corrected typos, results remain the sam

Compressed zero-divisor graphs of noncommutative rings

We extend the notion of the compressed zero-divisor graph $\varTheta(R)$ to noncommutative rings in a way that still induces a product preserving functor $\varTheta$ from the category of finite unital rings to the category of directed graphs. For a finite field $F$, we investigate the properties of $\varTheta(M_n(F))$, the graph of the matrix ring over $F$, and give a purely graph-theoretic characterization of this graph when $n \neq 3$. For $n \neq 2$ we prove that every graph automorphism of $\varTheta(M_n(F))$ is induced by a ring automorphism of $M_n(F)$. We also show that for finite unital rings $R$ and $S$, where $S$ is semisimple and has no homomorphic image isomorphic to a field, if $\varTheta(R) \cong \varTheta(S)$, then $R \cong S$. In particular, this holds if $S=M_n(F)$ with $n \neq 1$.Comment: 30 page

Categorial properties of compressed zero-divisor graphs of finite commutative rings

We define a compressed zero-divisor graph $\varTheta(K)$ of a finite commutative unital ring $K$, where the compression is performed by means of the associatedness relation. We prove that this is the best possible compression which induces a functor $\varTheta$, and that this functor preserves categorial products (in both directions). We use the structure of $\varTheta(K)$ to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.Comment: 14 page

The total zero-divisor graph of commutative rings

In this paper we initiate the study of the total zero-divisor graphs over commutative rings with unity. These graphs are constructed by both relations that arise from the zero-divisor graph and from the total graph of a ring. We characterize Artinian rings with the connected total zero-divisor graphs and give their diameters. Moreover, we compute major characteristics of the total zero-divisor graphs of the ring ${\mathbb Z}_m$ of integers modulo $m$ and prove that the total zero-divisor graphs of ${\mathbb Z}_m$ and ${\mathbb Z}_n$ are isomorphic if and only if $m=n$

A full scale Sklar's theorem in the imprecise setting

In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main tools we develop in order to do so are: (1) a theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) $p$-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) $p$-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladi\v{c} and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) $p$-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual $\sigma$-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.Comment: 16 pages, minor change

Coherence and avoidance of sure loss for standardized functions and semicopulas

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value $1$ at $(1,1,\ldots, 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A\leq C\leq B$ for standardized $n$-variate functions $A,B$ and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A\leq C\leq B$.Comment: 31 pages, 2 figure

Nilkolobarji in prakolobarji

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.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

Optimal real number graph labelings of a

subfamily of Kneser graphs