# Composition and orthogonality of derivations with multilinear polynomials in prime rings

• Balchand Prajapati
• Charu Gupta
Article

## Abstract

Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and $$\delta$$ be two derivations of R and S be the set of evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ over C which is not central valued. Let $$p,q\in R$$. We prove the followings.
1. (1)

If $$pud\delta (u)+\delta d(u)uq=0$$ for all $$u\in S$$ and $$p+q\notin C$$. Then either $$d=0$$ or $$\delta =0$$.

2. (2)

If $$pud(u)+d(u)uq=0$$ for all $$u\in S$$. Then either $$d=0$$ or $$p=q\in C$$, $$d(x)=[a,x]$$ for some $$a\in U$$ and $$f(x_1,\ldots ,x_n)^2$$ is central valued.

## Keywords

Prime ring Derivation Orthogonal derivations Extended centroid Utumi quotient ring

16W25 16N60

## Notes

### Acknowledgements

The authors are highly thankful to the referee for his/her several useful suggestions. First author is partially supported by the research Grant DST-SERB EMR/2016/001550.

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## Authors and Affiliations

• Balchand Prajapati
• 1
• Charu Gupta
• 1
1. 1.School of Liberal StudiesAmbedkar UniversityDelhiIndia