Linear orders on rings

The general ideas introduced in [8] are adopted in order to investigate the quasi cones and the cones of a ring. Using the finite extension property for cones, we answer the question, when a partial order in a partially ordered ring has a compatible linear extension (equivalently, when the positive cone is contained in a full cone). It turns out, that if there is no such extension, then it is caused by a finite system of polynomial like equations satisfied by some elements of a certain finite subset of the ring and some positive elements. As a consequence, we prove that a partial order can be linearly extended if and only if it can be linearly extended on every finitely generated subring

Cayley-Hamilton theorem for matrices over an arbitrary ring

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A

Maximal compatible extensions of partial orders

We give a complete description of maximal compatible partial orders on the mono-unary algebra $(A,f)$, where $f:A\to A$ is an arbitrary unary operation

Protestáns kisegyházak a 20. századi magyar társadalomban = Free Churches in Hungarian Society in the 20th Century

A pályázatban vállalt feladatainkat teljesítettük. Egy tankönyvnek is alkalmas kettősmonográfiával (Rajki Zoltán – Szigeti Jenő: Szabadegyházak története Magyarországon 1989-ig, Budapest, 2012, Gondolat) először tártuk fel a magyarországi szabadegyházak 1989 előtti történetét. Ezen kívül elkészült két monográfia egy-egy szabadegyházi közösség történetének egy-egy szakaszáról (Rajki Zoltán: A pünkösdi mozgalom története Magyarországon 1945 és 1961 között, Budapest, Gondolat, 2011; Rajki Zoltán: Az Egervári-mozgalom. A Keresztény Advent Közösség kialakulása és vallásszabadsági küzdelmei a Kádár-korszak második felében 1975-1990, Budapest, 2012, Gondolat). A 2011. májusi konferencia írásos anyaga megjelenés előtt áll a Gondolat Kiadónál. A fentieken kívül elkészítettünk egy online tanulmánykötetet. Több konferencián témával kapcsolatos előadásokat tartottunk. A fent említett könyveken, szerkesztett köteteken kívül 5 könyvfejezetet, 12 folyóiratcikket, 6 konferenciacikket publikáltunk az OTKA támogatásának feltüntetésével. Összeállítottunk egy tudományos weblapot (www.kisegyhazkutato.hu), amelyen számos forrást, folyóiratanyagot és 180 szabadegyházakra vonatkozó térképet tettünk közzé, hogy mások kutatómunkáját segítsük. | We have fulfilled all the OTKA grant expectations of the “Small Protestant Churches in the 20th Century Hungarian Society”. We finished and published the book: The History of the Hungarian Free Churches Before 1989 (Budapest, 2012, Gondolat) that is also suitable as a university textbook. In addition to this book, I also have published two monographs about different periods of the Seventh-day Adventist Church and the Pentecostal movement. (Rajki, Z.: A pünkösdi mozgalom története Magyarországon 1945 és 1961 között, Budapest, Gondolat, 2011; and Rajki, Z.: Az Egervári-mozgalom. A Keresztény Advent Közösség kialakulása és vallásszabadsági küzdelmei a Kádár-korszak második felében 1975-1990, Budapest, 2012, Gondolat). The written version of the “Free Churches, Religious Minorities and the Dictatorship in Europe in the 20th Centuries” conference held in May of 2011 will be published by the Gondolat Publishing House at the end of 2012. Furthermore, Jenő Szigeti and I have published a volume of studies in online form and we also gave presentations about our research topics in several conferences. Aside from the above mentioned books, we have published 5 book chapters and 13 articles in different journals and 5 conference papers which were supported by OTKA. We also have compiled a scholarly website (www.kisegyhazkutato.hu), with numerous sources, journals, and 180 maps about the Free Churches and made it available for free public use

Solving equations in the symmetric group

We investigate the solutions of the conjugate equations aya^(-1)=y^2 and aya^(-1)=y^(-2) in the symmetric group S_{n}. Here a is a fixed (constant) and y is a single unknown permutation (in S_{n}). It turns out that the existence of a non-trivial solution y heavily depends on the type of a

"A királyt tiszteljük, annak parancsolatait teljesítjük" : kommunisták voltak-e a hívők a dualizmus idején?

The first written “foot-prints” of the Pest-Buda believers came from the Wiener report dated in April 18, 1850. They were looking for a communist namedFerenc Szór, – who was never found –, however congregations gathered at private residence were reported to the localgovernment. Evidence proved that these local worhips were notdirected toward overthrowing the government. Although, State Government encouraged large churches to dissolve these sectarian gatherings, these initiatives had little or noresults. They helplessness were compensated with excuses like: because these bible groups are helping each other – they are probable communist people. Based on the existing sources, these documents reflects the relationship between the State and the just starting Nazaranian and Baptist groups, and reflects to the conflicts and their views to Local and State Government