On (anti-)multiplicative generalized derivations

Let R be a semiprime ring and let F, f : R → R be (not necessarily additive) maps satisfying F(xy)=F(x)y+xf(y) for all x,y R. Suppose that there are integers m and n such that F(uv)=mF(u)F(v)+nF(v)F(u) for all u, v in some nonzero ideal I of R. Under some mild assumptions on R, we prove that there exists c C(I⊥⊥) such that c=(m+n)c2, nc[I⊥⊥, I⊥⊥]=0 and F(x)=cx for all x I⊥⊥. The main result is then applied to the case when F is multiplicative or anti-multiplicative on I

O (anti)multiplikativnih posplošenih odvajanjih

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 ▫$I$▫.Let ▫$R$▫ be a semiprime ring and let ▫$F, f colon R to R$▫ be (not necessarily additive) maps satisfying ▫$F(xy) = F(x)y + xf(y)$▫ for all ▫$x,y in R$▫. Suppose that there are integers ▫$m$▫ and ▫$n$▫ such that ▫$F(uv) = mF(u)F(v) + nF(v)F(u)$▫ for all ▫$u$▫, ▫$v$▫ in some nonzero ideal ▫$I$▫ of ▫$R$▫. Under some mild assumptions on ▫$R$▫, we prove that there exists ▫$c in C(I^{botbot})$▫ such that ▫$c = (m+n)c^2$▫, ▫$nc[I^{botbot}, I^{botbot}] = 0$▫ and ▫$F(x) = cx$▫ for all ▫$x in I^{botbot}$▫. The main result is then applied to the case when ▫$F$▫ is multiplicative or anti-multiplicative on ▫$I$▫

Multiplikativna Liejeva n-odvajanja trikotnih kolobarjev

V članku je vpeljan pojem multiplikativnega Liejevega ▫$n$▫-odvajanja, ki predstavlja posplošitev pojma Liejevega (trojnega) odvajanja. Članek obravnava vprašanje, kdaj imajo vsa multiplikativna Liejeva ▫$n$▫-odvajanja trikotnega kolobarja ▫$mathcal{T}$▫ t.i. standardno obliko. Glavni rezultat se uporabi na klasičnih primerih trikotnih kolobarjev, kot so: gnezdne algebre in kolobarji (bločno) zgoraj trikotnih matrik.We introduce the notion of a multiplicative Lie ▫$n$▫-derivation of a ring, generalizing the notion of a Lie (triple) derivation. The main goal of the paper is to consider the question of when do all multiplicative Lie ▫$n$▫-derivations of a triangular ring ▫$mathcal{T}$▫ have the so-called standard form. The main result is applied to the classical examples of triangular rings: nest algebras and (block) upper triangular matrix rings

A characterization of the centroid of a prime ring

We characterize certain maps by their action on a fixed polynomial in noncommuting variables on algebras satisfying certain d -freeness condition. Consequently, a characterization of the centroid of a prime ring is obtained

Generalized Jordan derivations associated with Hochschild 2-cocycles of triangular algebras

summary:In this paper, we investigate a new type of generalized derivations associated with Hochschild 2-cocycles which is introduced by A.Nakajima (Turk.\ J.\ Math.\ 30 (2006), 403--411). We show that if $\mathcal U$ is a triangular algebra, then every generalized Jordan derivation of above type from $\mathcal U$ into itself is a generalized derivation