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

Development Permit System Policy Development in Ontario: Factors that Contribute to Implementation

In 2007, the Province of Ontario introduced a new planning tool called the Development Permit System (DPS) in order to increase the efficiency, certainty, and scope of the planning system. Despite these stated benefits, it has only been implemented in four municipalities. This paper examines the factors that could be limiting the implementation of the system in some municipalities, while encouraging implementation in others, using case studies conducted in five municipalities: the Township of Lake of Bays, the Township of Mulmur, the City of Kitchener, the Town of East Gwillimbury, and the City of Hamilton. Interviews were conducted with a municipal planner in each municipality to better understand how it came to the decision of whether or not to implement the DPS and the factors that contributed to this decision. The findings reveal that there are many factors that could be impacting a municipality’s decision to implement the system, such as the presence of a policy sponsor, the existence of a unique planning problem, satisfaction with the current planning system, and a general lack of knowledge of the DPS

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 $m=n$

REDČENJE PLODIČEV JABLANE SORT GALA IN ZLATI DELIŠES Z BA, NAA IN NJUNIMI KOMBINACIJAMI

Apple trees, eight-year-old ‘Gala’/M.9 and four-year-old ‘Golden Delicious’/M.9 have been thinned with 6- benzyladenine (BA) 50 ppm, 100 ppm and 200 ppm, with 1-naphthaleneacetic acid (NAA) 5 ppm, 10 ppm and 20 ppm, and with the tank mix combinations of BA 50 ppm + NAA 5 ppm or BA 20 ppm + NAA 5 ppm, all at 9-10 mm fruitlet diameter. All applied concentrations of BA and NAA thinned both cultivars significantly and no significant difference was found between BA or NAA thinning action. No concentration response on thinning was observed with neither BA nor NAA application. All BA or NAA separate treatments caused yield of fruit to shift to bigger size class. The only concentration response effect was found on evaluation of mean fruit weight data on ‘Golden Delicious’. The higher concentration of BA was used, the higher was the weight of ‘Golden Delicious’ fruit. On the opposite, the higher concentration of NAA was used, the lower was the weight of ‘Golden Delicious’ fruit (not significantly). When BA and NAA were sprayed on ‘Gala’ or ‘Golden Delicious’ as a tank mix combination, similar effect on thinning or fruit growth occurred comparing to BA or NAA when sprayed alone. Return bloom was enhanced on all thinned ‘Gala’ trees while flower formation of ‘Golden Delicious’ was better in the case of BA 200 ppm, NAA 5 ppm, NAA 20 ppm or tank mix spraying of BA 20 ppm + NAA 5 ppm.Sedem let stare jablane sorte Gala/M.9 in štiri leta stare jablane sorte Zlati delišes/M.9 smo redčili s 6- benziladeninom (BA) v koncentraciji 50 ppm, 100 ppm in 200 ppm, nadalje z 1-naftilocetno kislino (NAA) v koncentraciji 5 ppm, 10 ppm in 20 ppm, ter s kombinacijo obeh sredstev, to je z mešanico BA 50 ppm + NAA 5 ppm, ter BA 20 ppm + NAA 5 ppm. Sredstva za redčenje smo nanesli v času poprečnega premera plodičev 9-10 mm. BA in NAA sta redčili obe preskušani sorti značilno in s približno isto stopnjo trebljenja plodičev. Trebljenje plodičev po nanosu različnih koncentracij BA oziroma NAA ni bilo odvisno od uporabljenih koncentracij. Pridelek plodov večjega velikostnega razreda se je pri obeh sortah povečal pri vseh nanosih BA ali NAA. Pri sorti Zlati delišes smo opazili učinek koncentracij uporabljenega sredstva na poprečno težo plodov. Po nanosu večje koncentracije BA je bila tudi poprečna teža plodov sorte Zlati delišes večja. Obratno pa je nanos večje koncentracije NAA povzročil manjšo poprečno težo plodov (statistično neznačilno) sorte Zlati delišes. V primeru škropljenja mešanice BA + NAA, sta obe obravnavani sorti reagirali z enako intenziteto trebljenja plodičev, kot v primeru samostojnega nanosa enega od obeh sredstev. Povratno cvetenje je bilo značilno boljše pri vseh redčenih drevesih sorte Gala, pri sorti Zlati delišes pa se je nastavek cvetja izboljšal le po nanosih BA 200 ppm, NAA 5 ppm, NAA 20 ppm in pri mešanici BA 20 ppm + NAA 5 ppm