1. On certain functional equation in prime ringsMaja Fošner, Benjamin Marcen, Joso Vukman, 2022, original scientific article Abstract: The purpose of this paper is to prove the following result. Let R be prime ring of characteristic different from two and three, and let F:R→R be an additive mapping satisfying the relation F(x3)=F(x2)x−xF(x)x+xF(x2) for all x∈R. In this case, F is of the form 4F(x)=D(x)+qx+xq for all x∈R, where D:R→R is a derivation, and q is some fixed element from the symmetric Martindale ring of quotients of R.
Keywords: prime ring, derivation, Jordan derivation, functional equation, algebra
Published in DKUM: 12.06.2024; Views: 131; Downloads: 16
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2. On functional equations related to derivations in semiprime rings and standard operator algebrasNejc Širovnik, 2012, original scientific article Abstract: In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let ▫$X$▫ be a real or complex Banach space, let ▫$L(X)$▫ be the algebra of all bounded linear operators of ▫$X$▫ into itself and let ▫$A(X) subset L(X)$▫ be a standard operator algebra. Suppose there exist linear mappings ▫$D,G colon A(X) to L(X)$▫ satisfying the relations ▫$D(A^3)=D(A^2)A + A^2G(A)$▫, ▫$G(A^3) = G(A^2)A + A^2D(A)$▫ for all ▫$A in A(X)$▫. In this case there exists ▫$B in L(X)$▫ such that ▫$D(A) = G(A) = [A,B]$▫ holds for all ▫$A in A(X)$▫.
Keywords: matematika, algebra, prakolobar, polprakolobar, Banachov prostor, standardna operatorska algebra, odvajanje, jordansko odvajanje, jordansko trojno odvajanje, mathematics, algebra, prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation
Published in DKUM: 10.07.2015; Views: 1479; Downloads: 68
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3. On some functional equations arising from (m, n)-Jordan derivations and commutativity of prime ringsMaja Fošner, Joso Vukman, 2012, original scientific article Abstract: The purpose of this paper is to prove the following result. Let ▫$m, n ge 1$▫ be some fixed integers with ▫$m ne n$▫, and let ▫$R$▫ be a prime ring with ▫$(m+n)^2 < text{char} (R)$▫. Suppose a nonzero additive mapping ▫$D : R to R$▫ exists satisfying the relation ▫$(m+n)^2 D(x^3) = m(3m+n) D(x)x^2 + 4mnxD(x)x + n(3n+m)x^2 D(x)$▫ for all ▫$x in R$▫. In this case ▫$D$▫ is a derivation and ▫$R$▫ is commutative.
Keywords: matematika, prakolobar, polprakolobar, odvajanje, jordansko odvajanje, levo odvajanje, mathematics, prime ring, semiprime ring, derivation, Jordan derivation, left dderivation, left Jordan derivation, (m, n)-Jordan drivation
Published in DKUM: 10.07.2015; Views: 1292; Downloads: 88
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4. On certain functional equation arising from (m, n)-Jordan centralizers in prime ringsNina Peršin, Joso Vukman, 2012, original scientific article Abstract: The purpose of this paper is to prove the following result. Let ▫$m ge 1$▫, ▫$n ge 1$▫ be some fixed integers and let ▫$R$▫ be a prime ring with ▫$text{char}(R)= 0$▫ or ▫$(m+n)^2 < text{char}(R)$▫. Suppose there exists an additive mapping ▫$T colon R to R$▫ satisfying the relation ▫$2(m+n)^2T(x^3) = m(2m+n)T(x)x^2 + 2mnxT(x)x + n(2n+m)x^2T(x)$▫ for all ▫$x in R$▫. In this case ▫$T$▫ is a two-sided centralizer.
Keywords: matematika, algebra, kolobar, prakolobar, polprakolobar, Banachov prostor, Hilbertov prostor, algebra vseh omejenih linearnih operatorjev, standardna operatorska algebra, odvajanje, jordansko odvajanje, centralizator, algebra, ring, prime ring, semiprime ring, Banach space, Hilbert space, algebra of all bounded linear operators, standard operator algebra, derivation, Jordan derivation, left (right) centralizer, two-sided centralizer, left (right) Jordan centralizer, (m, n)-Jordan centralizer
Published in DKUM: 10.07.2015; Views: 1422; Downloads: 140
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5. Identities with generalized derivations in prime ringsMaja Fošner, Joso Vukman, 2012, original scientific article Abstract: In this paper we investigate identities with two generalized derivations in prime rings. We prove, for example, the following result. Let ▫$R$▫ be a prime ring of characteristic different from two and let ▫$F_1, F_2 colon R to R$▫ be generalized derivations satisfying the relation ▫$F_1(x)F_2(x) + F_2(x)F_1(x) = 0$▫ for all ▫$x in R$▫. In this case either ▫$F_1 = 0$▫ or ▫$F_2 = 0$▫.
Keywords: matematika, prakolobar, polprakolobar, odvajanje, posplošeno odvajanje, mathematics, prime ring, semiprime ring, derivation, generalized derivation
Published in DKUM: 10.07.2015; Views: 1212; Downloads: 100
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6. An equation related to two-sided centralizers in prime ringsMaja Fošner, Joso Vukman, 2011, original scientific article Abstract: The purpose of this paper is to prove the following result. Let ▫$m$▫ and ▫$n$▫ be positive integers, and let ▫$R$▫ be a prime ring with char▫$(R)=0$▫ or ▫$m+n+1 le char(R)$▫. Let ▫$T colon R to R$▫ be an additive mapping satisfying the relation ▫$T(x^{m+n+1}) = {x^m}T(x)x^n$▫ for all ▫$x in R$▫. In this case ▫$T$▫ is a two-sided centralizer.
Keywords: matematika, algebra, prakolobar, funkcijska identiteta, dvostranski centralizator, mathematics, prime ring, functional identity, two-sided centralizer
Published in DKUM: 10.07.2015; Views: 1102; Downloads: 94
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7. Some remarks on derivations in semiprime rings and standard operator algebrasJoso Vukman, 2011, original scientific article Abstract: Identities related to derivations on semiprime rings and standard operator algebras are investigated. We prove the following result which generalizes a classical result of Chernoff. Let ▫$X$▫ be a real or complex Banach space, let ▫$L(X)$▫ be the algebra of all bounded linear operators of ▫$X$▫ into itself and let ▫$A(X) subseteq L(X)$▫ be a standard operator algebra. Suppose there exists a linear mapping ▫$D:A(X) to L(X)$▫ satisfying the relation ▫$2D(A^{3}) = D(A^2)A + A^2D(A) + D(A)A^2 + AD(A^2)$▫ for all ▫$A in A(X)$▫. In this case ▫$D$▫ is of the form ▫$D(A) = AB-BA$▫ for all ▫$A in A(X)$▫ and some fixed ▫$B in L(X)$▫, which means that ▫$D$▫ is a linear derivation.
Keywords: matematika, algebra, prakolobar, polprakolobar, Banachov prostor, standardna operatorska algebra, odvajanje, jordansko odvajanje, jordansko trojno odvajanje, mathematics, algebra, prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation
Published in DKUM: 10.07.2015; Views: 1598; Downloads: 131
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8. Equations related to derivations on prime ringsMaja Fošner, Joso Vukman, 2011, original scientific article Abstract: In this paper we prove the following result. Let ▫$m ge 0$▫ and ▫$nge 0$▫ be integers with ▫$m+n ne 0$▫ and let ▫$R$▫ be a prime ring with ▫$char(R)=0$▫ or ▫$m+n+1 le char(R) ne 2$▫. Suppose there exists a nonzero additive mapping ▫$D:R to R$▫ satisfying the relation ▫$D(x^{m+n+1}) = (m+n+1)x^m D(x)x^n$▫ for all ▫$x in R$▫. In this case ▫$D$▫ is a derivation and ▫$R$▫ is commutative.
Keywords: matematika, prakolobar, funkcijska identiteta, odvajanje, mathematics, prime ring, functional identity, derivation
Published in DKUM: 10.07.2015; Views: 1181; Downloads: 124
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9. On (anti-)multiplicative generalized derivationsDaniel Eremita, Dijana Ilišević, 2012, original scientific article Abstract: Naj bo ▫$R$▫ polprakolobar in naj bosta ▫$F, f colon R to R$▫ taki (ne nujno aditivni) preslikavi, da je ▫$F(xy) = F(x)y + xf(y)$▫ za vse ▫$x,y in R$▫. Denimo, da obstajata taki celi števili ▫$m$▫ in ▫$n$▫, da velja ▫$F(uv) = mF(u)F(v) + nF(v)F(u)$▫ za vse elemente ▫$u$▫, ▫$v$▫ neničelnega ideala ▫$I$▫ kolobarja ▫$R$▫. Ob določenih blagih predpostavkah za polprakolobar ▫$R$▫ dokažemo, da obstaja tak ▫$c in C(I^{botbot})$▫, da je ▫$c = (m+n)c^2$▫, ▫$nc[I^{botbot}, I^{botbot}] = 0$▫ in ▫$F(x) = cx$▫ za vse ▫$x in I^{botbot}$▫. Glavni rezultat je nato uporabljen v primeru, ko je preslikava ▫$F$▫ multiplikativna ali antimultiplikativna na idealu ▫$I$▫.
Keywords: matematika, algebra, aditivnost, kolobar, prakolobar, polprakolobar, odvajanje, posplošeno odvajanje, homomorfizem, antihomomorfizem, algebra, additivity, ring, prime ring, semiprime ring, derivation, generalized derivation, homomorphism, anti-homomorphism
Published in DKUM: 10.07.2015; Views: 2042; Downloads: 142
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10. On some equations in prime ringsMaja Fošner, Joso Vukman, 2007, original scientific article Abstract: The main purpose of this paper is to prove the following result. Let ▫$R$▫ be a prime ring of characteristic different from two and let ▫$T : R to R$▫ be an additive mapping satisfying the relation ▫$T(x^3) = T(x)x^{2} - xT(x)x + x^{2} T(x)$▫ for all ▫$x in R$▫. In this case ▫$T$▫ is of the form ▫$4T(x) = qx + xq$▫, where ▫$q$▫ is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.
Keywords: matematika, algebra, prakolobar, polprakolobar, funkcijska identiteta, odvajanje, jordansko odvajanje, involucija, bicirkularni projektor, mathematics, algebra, prime ring, semiprime ring, functional identity, derivation, Jordan derivation, involution, bicircular projection
Published in DKUM: 10.07.2015; Views: 1398; Downloads: 85
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