On functional equations related to bicircular projections
Vukman, Joso
In this paper we prove the following result. Let ▫$R$▫ be a 2-torsion free semiprime ▫$\ast$▫-ring. Suppose that ▫$D, G : R \to R$▫ are additive mappings satisfying the relations ...▫$D(xyx)=D(x)yx+xG(y^{\ast})^{\ast}x+xyD(x), G(xyx)= G(x)yx+xD(y^{\ast})^{ast}x+xyG(x)$▫ for all pairs ▫$x,y \in R$▫. In which case ▫$D$▫ and ▫$G$▫ are of the form ▫$8D(x)=2(d(x)+g(x))+(p+q)x+x(p+q), 8G(x)=2(d(x)-g(x))+(q-p)x+x(q-p)$▫, for all ▫$x \in R$▫, where d, g are derivations of ▫$R$▫ and p, q are some elements form symmetric Martindale ring of quotients of ▫$R$▫. Besides, ▫$d(x)=-d(x^{\ast})^{\ast}, g(x)=g(x^{\ast})^{\ast}$▫, for all ▫$x \in R$▫, and ▫$p^{\ast} = p,q^{\ast} = -q$▫.
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