Some identities involving multiplicative semiderivations on ideals

Oznur GOLBASİ; Zeliha BEDİR

doi:https://doi.org/10.15672/hujms.650600

Article Details

Some identities involving multiplicative semiderivations on ideals

Authors:

Oznur GOLBASİ | Zeliha BEDİR

Publisher:

Hacettepe University

Main Topics:

Mathematics

Journal:

Hacettepe Journal of Mathematics and Statistics 2021 Vol.50 Issue 4, pp.963-969

Document Type:

Article

Publication Date:

8/2021

Document Language:

English

Document ID (Org No.):

2000450551

Frequency:

Bimonthly

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.650600

Abstract:

 

Let RRR be a prime ring and III be a nonzero ideal of R.R.R. A mapping d:R→Rd:R→Rd:R\rightarrow R is called a multiplicative semiderivation if there exists a function g:R→Rg:R→Rg:R\rightarrow R such that (i) d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y) and (ii) d(g(x))=g(d(x))d(g(x))=g(d(x))d(g(x))=g(d(x)) hold for all x,y∈R.x,y∈R.x,y\in R. In the present paper, we shall prove that [x,d(x)]=0,[x,d(x)]=0,[x,d(x)]=0, for all x∈Ix∈Ix\in I if any of the followings holds: i) d(xy)±xy∈Z,d(xy)±xy∈Z,d(xy)\pm xy\in Z, ii) d(xy)±yx∈Z,d(xy)±yx∈Z,d(xy)\pm yx\in Z, iii) d(x)d(y)±xy∈Z,d(x)d(y)±xy∈Z,d(x)d(y)\pm xy\in Z, iv) d(xy)±d(x)d(y)∈Z,d(xy)±d(x)d(y)∈Z,d(xy)\pm d(x)d(y)\in Z, viii) d(xy)±d(y)d(x)∈Z,d(xy)±d(y)d(x)∈Z,d(xy)\pm d(y)d(x)\in Z, for all x,y∈I.x,y∈I.x,y\in I. Also, we show that RRR must be commutative if d(I)⊆Z.d(I)⊆Z.d(I)\subseteq Z.

Article Details

Some identities involving multiplicative semiderivations on ideals

Authors: Oznur GOLBASİ | Zeliha BEDİR

Publisher: Hacettepe University

Main Topics: Mathematics

Journal: Hacettepe Journal of Mathematics and Statistics 2021 Vol.50 Issue 4, pp.963-969

Document Type: Article

Publication Date: 8/2021

Document Language: English

Document ID (Org No.): 2000450551

Frequency: Bimonthly

eISSN: 2651-477X

pISSN: 2651-477X

DOI: https://doi.org/10.15672/hujms.650600

Issues

Authors:

Oznur GOLBASİ | Zeliha BEDİR

Publisher:

Hacettepe University

Main Topics:

Mathematics

Journal:

Hacettepe Journal of Mathematics and Statistics 2021 Vol.50 Issue 4, pp.963-969

Document Type:

Article

Publication Date:

8/2021

Document Language:

English

Document ID (Org No.):

2000450551

Frequency:

Bimonthly

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.650600