20.500.12556/DKUM-43059 Posebne funkcionalne enačbe na prakolobarjih Special functional equations on prime rings V doktorski disertaciji so obravnavane funkcionalne enačbe, ki so v zvezi z odvajanji, centralizatorji in sorodnimi preslikavami na prakolobarjih. Med slovenskimi matematiki se je s tem področjem matematike v osemdesetih letih prejšnjega stoletja začel prvi ukvarjati J. Vukman, sledili so M. Brešar, B. Zalar, B. Hvala in v novejšem času M. Fošner, D. Benkovič, D. Eremita, I. Kosi-Ulbl in A. Fošner. Osnovno sredstvo pri reševanju tovrstnih funkcionalnih enačb je uporaba teorije funkcijskih identitet. Nekoliko natančneje pojasnimo omenjene pojme. Aditivna preslikava D, ki slika poljuben kolobar R vase, je odvajanje, če velja D(xy) = D(x)y + xD(y) za vsak par x, y iz R in je jordansko odvajanje, če velja D(x^2)=D(x)x +xD(x). Očitno je, da je vsako odvajanje tudi jordansko odvajanje, obratno pa v splošnem ne velja. I. N. Herstein je leta 1957 dokazal, da je vsako jordansko odvajanje na prakolobarju s karakteristiko različno od dva, odvajanje. V doktorski disertaciji se najprej osredotočimo na funkcionalne enačbe, ki so v zvezi z odvajanji. Obravnavali smo funkcionalni enačbi D(x^3=D(x^2)x + x^2D(x) in D(x^3=D(x)x^2+ xD(x^2),kjer je D aditivna preslikava, ki slika prakolobar s primernimi omejitvami glede karakteristike vase. Dokazali smo, da je D odvajanje. Nadalje poiščemo tudi rešitev funkcionalne enačbe 2D(x^(m+n+1))=(m+n+1)(x^mD(x)x^n+x^nD(x)x^m), kjer sta m in n fiksni naravni števili in D neničelna aditivna preslikava, ki slika prakolobar s primernimi omejitvami glede karakteristike vase. Dokažemo, da je D odvajanje in R komutativen kolobar. V tretjem poglavju so obravnavane funkcionalne enačbe, ki so v zvezi s centralizatorji. Aditivna preslikava T, ki slika poljuben kolobar R vase, je levi (desni) centralizator, če je T(xy)=T(x)y (T(xy)=xT(y)) za vsak par x, y iz R. V prvem podpoglavju tega razdelka je obravnavana funkcionalna enačba 2T(x^(m+n+1))=x^mT(x)x^n +x^nT(x)x^m na prakolobarju s primernimi omejitvami glede karakteristike, kjer sta sta m in n fiksni nenegativni celi števili in m+n je različno od 0. Dokažemo, da je T dvostranski centralizator. Aditivna preslikava T, ki slika poljuben kolobar R vase, je (m,n)-jordanski centralizator, če je (m+n)T(x^2)=mT(x)x+nxT(x) za vsak x iz R, kjer sta m in n fiksni nenegativni celi števili in m+n je različno od 0. Ta pojem je leta 2010 vpeljal J. Vukman ter med drugim tudi dokazal, da vsak (m,n)-jordanski centralizator na poljubnem kolobarju R zadošča pogoju 2(m+n)^2T(xyx) = mnT(x)xy + m(2m + n)T(x)yx -mnT(y)x^2 + 2mnxT(y)x - mnx^2T(y) + n(m + 2n)xyT(x) + mnyxT(x) za vsak par x, y iz R. Če v tej identiteti piŠemo y = x, dobimo naslednjo funkcionalno enačbo 2(m+n)^2T(x3)=m(2m+n)T(x)x^2+2mnxT(x)x+n(m+2n)x^2T(x), ki je obravnavana v zadnjem delu doktorske disertacije na prakolobarju s primernimi omejitvami glede karakteristike, kjer sta m in n fiksni naravni števili. Dokažemo, da je T dvostranski centralizator. V zaključnem poglavju podamo odprta vprašanja o funkcionalnih enačbah, ki so v zvezi s posplošenimi odvajanji in (theta, phi)- odvajanji, kjer sta theta in phi avtomorfzma na kolobarju R. Special functional equations on prime rings The central object in this thesis are functional equations related to derivations, centralizers and other similary mappings on prime rings. Slovenian mathematicians who first dealt with similar mathers in 80's of the previous century, were J. Vukman, M. Brešar, B. Zalar, B. Hvala and recently also M. Fošner, D. Benkovič, D. Eremita, I. Kosi-Ulbl and A. Fošner. The basic tool has been the theory of functional identities. Let us look into these matters more closely. An additive mapping D which maps an arbitrary ring R into itself, is called a derivation if D(xy)=D(x)y + xD(y) holds for all pairs x,y in R and is called a Jordan derivation in case D(x^2)=D(x)x + xD(x). It is obvious that every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein asserts that any Jordan derivation on a prime ring with char(R) diferent from 2 is a derivation. In the first part we focus on functional equations connected with derivations. We deal with functional equations D(x^3=D(x^2)x + x^2D(x) and D(x^3)=D(x)x^2+ xD(x^2), where D is an additive mapping, which maps a ring R, with suitable characteristic restrictions, into itself. In this case D is a derivation. Furthermore, we have found a solution of functional equation 2D(x^(m+n+1))=(m+n+1)(x^mD(x)x^n + x^nD(x)x^m), where m and n are some fixed integers and D is an additive nonzero mapping, which maps ring R, with suitable characteristic restrictions, into itself. In this case D is a derivation and R is commutative ring. Third chapter is about functional equations related to centralizers. An additive mapping T which maps an arbitrary ring R into itself, is called a left (right) centralizer in case T(xy)=T(x)y (T(xy) = xT(y)) holds for all pairs x, y in R. In the first subsection we introduce the solution of the functional equation 2T(x^(m+n+1))=x^mT(x)x^n + x^nT(x)x^m on a prime ring with suitable characteristic restrictions, where m>0 and n>0 are some fixed integers and m+n is not 0. In this case T is a two-sided centralizer. An additive mapping T which maps an arbitrary ring R into itself, is called an (m,n)-Jordan centralizer in case (m+n)T(x^2)= mT(x)x+nxT(x) holds for all x in R, where m, n are some fixed integers and m+n is not 0. The concept of (m,n)-Jordan centralizer was introduced by Vukman in 2010. In this article he also proved that any (m, n)-Jordan centralizer on an arbitrary ring R, satisfies the relation 2(m+n)2T(xyx) = mnT(x)xy + m(2m + n)T(x)yx - mnT(y)x^2 + 2mnxT(y)x - mnx^2T(y) + n(m + 2n)xyT(x) + mnyxT(x) for all pairs x, y in R. If in this identity we write y = x, we get another functional equation 2(m+ n)^2T(x^3)=m(2m+ n)T(x)x^2+ 2mnxT(x)x+n(m+ 2n)x^2T(x), which we treat in the very last part on a prime ring, with suitable charachteristic restrictions, where m, n are some fixed integers. We prove that in this case T is a two-sided centralizer. We conclude with some open questions on functional equations related to generalized derivations and (theta, phi)-derivations, where theta and phi are automorphisms of R. aditivna preslikava desni (levi) centralizator d-prosta množica dvostranski centralizator funkcijska identiteta jordansko odvajanje komutirajoča preslikava (m n)-jordanski centralizator odvajanje polprakolobar prakolobar standardna rešitev. Additive mapping commuting mapping derivation d-free sets functional identity Jordan derivation (m n)-Jordan centralizer prime ring right (left) centralizer semiprime ring standard solution two-sided centralizer true true false N. Peršin] Slovenski jezik Angleški jezik Doktorska disertacija 2013-11-21 19:34:56 2013-12-05 12:58:54 2022-05-09 03:05:33 2021-04-27 03:41:19 2013 0 [S. l. 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 512.552(043.3) 20201992 URN:SI:UM:DK:134N3RJF DR_Persin_Nina_2013.pdf DR_Persin_Nina_2013.pdf 1 8044C851D264A060B07B42B57C80486B c0669ce123bb70a80c9c09464004165320209623c8aedab2742b6e2d546f0dab 8bdff240-7c0a-11eb-bb7a-00155d0001ca https://dk.um.si/Dokument.php?lang=slv&id=60683 Fakulteta za naravoslovje in matematiko 0 0 0