#### Glasnik Matematicki, Vol. 41, No.1 (2006), 51-55.

### ON FUNCTIONAL EQUATIONS RELATED TO BICIRCULAR PROJECTIONS

### Joso Vukman

Department of Mathematics, University of Maribor, PEF, Koroška 160, 2000 Maribor, Slovenia

*e-mail:* `joso.vukman@uni-mb.si`

**Abstract.** In this paper we prove the following result. Let *R*
be a 2-torsion free semiprime *-ring. Suppose that
*D*, *G* : *R* → *R*
are additive mappings satisfying the relations

*D*(*xyx*) = *D*(*x*)*yx* + *xG*(*y**)**x* +
*xyD*(*x*),
*G*(*xyx*) = *G*(*x*)*yx* + *xD*(y*)**x*
+ *xyG*(*x*),

for all pairs *x*, *y* *R*.
In this case *D* and *G*
are of the form
8*D*(*x*) = 2(*d*(*x*) + *g*(*x*)) +
(*p* + *q*)*x* + *x*(*p* + *q*),
8*G*(*x*) = 2(*d*(*x*) - *g*(*x*)) +
(*q* - *p*)*x* + *x*(*q* - *p*),

for all *x* *R*, where *d*, *g*
are derivations of *R* and *p*, *q* are some elements from
symmetric Martindale ring of quotients of *R*.
Besides, *d*(*x*) = -*d*(*x**)*,
*g*(*x*) = *g*(*x**)*,
for all *x* *R*, and *p** = *p*,
*q** = -*q*.
**2000 Mathematics Subject Classification.**
16E99.

**Key words and phrases.** *-ring, semiprime ring, derivation,
left (right) centralizer, bicircular projection.

**Full text (PDF)** (free access)
DOI: 10.3336/gm.41.1.04

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*Bicircular projections and
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CrossRef

- J. Vukman, I. Kosi-Ulbl and D. Eremita,
*On certain equations in rings*,
Bull. Austral. Math. Soc. **71** (2005), 53-60.

MathSciNet

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