Some identities involving multiplicative semiderivations on ideals

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Gölbaşı Ö., Bedir Z.

HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS, vol.50, no.4, pp.963-969, 2021 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 4
  • Publication Date: 2021
  • Doi Number: 10.15672/hujms.650600
  • Journal Name: HACETTEPE JOURNAL OF MATHEMATICS AND STATISTICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), TR DİZİN (ULAKBİM)
  • Page Numbers: pp.963-969
  • Keywords: prime rings, semiderivation, multiplicative semiderivation, DERIVATIONS, PRIME, MAPPINGS
  • Sivas Cumhuriyet University Affiliated: Yes

Abstract

Let R be a prime ring and I be a nonzero ideal of R. A mapping d : R -> R is called a multiplicative semiderivation if there exists a function g : R -> R such that (i) d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and (ii) d(g(x)) = g(d(x)) hold for all x, y is an element of R. In the present paper, we shall prove that [x, d(x)] = 0, for all x is an element of I if any of the followings holds: i) d(xy) +/- xy is an element of Z, H) d(xy) +/- yx is an element of Z, Hi) d(x)d(y) +/- xy is an element of Z, iv) d(xy) +/- d(x)d(y) is an element of Z, viii) d(xy) +/- d(y)d(x) is an element of Z, for all x, y is an element of I. Also, we show that R must be commutative if d(I) subset of Z.