Research Article | Open Access

Basudeb Dhara, Atanu Pattanayak, "On Generalized ()-Derivations in Semiprime Rings", *International Scholarly Research Notices*, vol. 2012, Article ID 120251, 7 pages, 2012. https://doi.org/10.5402/2012/120251

# On Generalized ()-Derivations in Semiprime Rings

**Academic Editor:**C. Munuera

#### Abstract

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

#### 1. Introduction

Throughout the present paper, always denotes an associative semiprime ring with center . For any , the commutator and anticommutator of and are denoted by and and are defined by and , respectively. Recall that a ring is said to be prime, if for , implies either or and is said to be semiprime if for , implies . An additive mapping is said to be derivation if holds for all . The notion of derivation is extended to generalized derivation. The generalized derivation means an additive mapping associated with a derivation such that holds for all . Then every derivation is a generalized derivation, but the converse is not true in general.

A number of authors have studied the commutativity theorems in prime and semiprime rings admitting derivation and generalized derivation (see e.g., [1–8]; where further references can be found).

Let and be two endomorphisms of . For any , set and . An additive mapping is called a -derivation if holds for all . By this definition, every -derivation is a derivation, where means the identity map of . In the same manner the concept of generalized derivation is also extended to generalized -derivation as follows. An additive map is called a generalized -derivation if there exists a -derivation such that holds for all . Of course every generalized -derivation is a generalized derivation of , where denotes the identity map of .

There is also ongoing interest to study the commutativity in prime and semiprime rings with -derivations or generalized -derivations (see [9–17]).

The present paper is motivated by the results of [17]. In [17], Rehman et al. have discussed the commutativity of a prime ring on generalized -derivation, where and are automorphisms of . More precisely, they studied the following situations: (i) ; (ii) ; (iii) ; (iv) ; (v) for all , where is a nonzero ideal of .

The main objective of the present paper is to extend above results for generalized -derivations in semiprime ring , where and are considered as epimorphisms of .

To prove our theorems, we will frequently use the following basic identities:

#### 2. Main Results

Theorem 2.1. * Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *First we consider the case
for all . Replacing by in (2.1) we get
Using (2.1), it reduces to
for all . Again replacing by in (2.3), we get
for all and . Left multiplying (2.3) by and then subtracting from (2.4) we have
for all and . Replacing with , , we get for all and . Since is an epimorphism of , we can write
for all .

Since is semiprime, it must contain a family of prime ideals such that . If is a typical member of and , it follows that

Construct two additive subgroups and . Then . Since a group cannot be a union of two its proper subgroups, either or , that is, either or . Thus both cases together yield for any . Therefore, , that is, . Thus
and so . This implies , where is a nonzero ideal of , since . Then . Since is semiprime, it follows that , that is, .

Similarly, we can obtain the same conclusion when for all .

Theorem 2.2. *Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *We begin with the case
for all . Replacing by in (2.9) we get
Right multiplying (2.9) by and then subtracting from (2.10) we get
for all .

Now replacing by in (2.11), we obtain
for all and for all . Left multiplying (2.11) by and then subtracting from (2.12), we get
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar results hold in case for all .

Theorem 2.3. *Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *We assume first that for all . This implies
Replacing by in (2.14) we have
Right multiplying (2.14) by and then subtracting from (2.15), we get
Now replacing by , where , in (2.16), we obtain
Left multiplying (2.16) by and then subtracting from (2.17), we get that
that is,
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar results hold in case for all .

Theorem 2.4.

* Proof. *By our assumption first consider for all . This gives
Replacing by in (2.20), we have
Right multiplying (2.20) by and then subtracting from (2.21), we obtain that
Now replacing by , where , in (2.22) and by using (2.22), we obtain
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

Similar argument can be adapted in case for all .

Theorem 2.5. *Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a nonzero -derivation of such that . If for all , then contains a nonzero central ideal. *

* Proof. *We begin with the situation
for all . Replacing by in (2.24), we get
Right multiplying (2.24) by and then subtracting from (2.25), we obtain that
for all . Now replacing by in (2.26), where , and by using (2.26), we obtain
for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.

In case for all , the similar argument can be adapted to draw the same conclusion.

We know the fact that if a prime ring contains a nonzero central ideal, then must be commutative (see Lemma 2 in [18]). Hence the following corollary is straightforward.

Corollary 2.6. *Let be a prime ring, and two epimorphisms of and a generalized -derivation associated with a nonzero -derivation of satisfying any one of the following conditions:*(1)* for all or for all ;*(2)* for all or for all ;*(3)* for all or for all ;*(4)* for all or for all ;*(5)* for all or for all ;**then must be commutative.*

#### References

- M. Ashraf, N. Rehman, S. Ali, and M. R. Mozumder, “On semiprime rings with generalized derivations,”
*Boletim da Sociedade Paranaense de Matemática. 3rd Série*, vol. 28, no. 2, pp. 25–32, 2010. View at: Publisher Site | Google Scholar - M. Ashraf, A. Ali, and S. Ali, “Some commutativity theorems for rings with generalized derivations,”
*Southeast Asian Bulletin of Mathematics*, vol. 31, no. 3, pp. 415–421, 2007. View at: Google Scholar - H. E. Bell and N.-U. Rehman, “Generalized derivations with commutativity and anti-commutativity conditions,”
*Mathematical Journal of Okayama University*, vol. 49, pp. 139–147, 2007. View at: Google Scholar - Q. Deng and H. E. Bell, “On derivations and commutativity in semiprime rings,”
*Communications in Algebra*, vol. 23, no. 10, pp. 3705–3713, 1995. View at: Publisher Site | Google Scholar - B. Dhara, “Remarks on generalized derivations in prime and semiprime rings,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 646587, 6 pages, 2010. View at: Publisher Site | Google Scholar - B. Hvala, “Generalized derivations in rings,”
*Communications in Algebra*, vol. 26, no. 4, pp. 1147–1166, 1998. View at: Publisher Site | Google Scholar - E. C. Posner, “Derivations in prime rings,”
*Proceedings of the American Mathematical Society*, vol. 8, pp. 1093–1100, 1957. View at: Google Scholar - M. A. Quadri, M. S. Khan, and N. Rehman, “Generalized derivations and commutativity of prime rings,”
*Indian Journal of Pure and Applied Mathematics*, vol. 34, no. 9, pp. 1393–1396, 2003. View at: Google Scholar - F. Ali and M. A. Chaudhry, “On generalized $(\alpha ,\beta )$-derivations of semiprime rings,”
*Turkish Journal of Mathematics*, vol. 35, no. 3, pp. 399–404, 2011. View at: Google Scholar - N. Argaç and E. Albas, “On generalized $(\sigma ,\tau )$-derivations,”
*Sibirskiĭ Matematicheskiĭ Zhurnal*, vol. 43, no. 6, pp. 977–984, 2002. View at: Publisher Site | Google Scholar - N. Argaç, A. Kaya, and A. Kisir, “$(\sigma ,\tau )$-derivations in prime rings,”
*Mathematical Journal of Okayama University*, vol. 29, pp. 173–177, 1987. View at: Google Scholar - N. Aydın and K. Kaya, “Some generalizations in prime rings with $(\sigma ,\tau )$-derivation,”
*Turkish Journal of Mathematics*, vol. 16, no. 3, pp. 169–176, 1992. View at: Google Scholar - ö. Gölbaşi and E. Koç, “Some commutativity theorems of prime rings with generalized $(\sigma ,\tau )$-derivation,”
*Communications of the Korean Mathematical Society*, vol. 26, no. 3, pp. 445–454, 2011. View at: Publisher Site | Google Scholar - E. Gölbaşi and E. Koç, “Notes on generalized $(\sigma ,\tau )$-derivation,”
*Rendiconti del Seminario Matematico della Università di Padova*, vol. 123, pp. 131–139, 2010. View at: Google Scholar - Y.-S. Jung and K.-H. Park, “On generalized $(\alpha ,\beta )$-derivations and commutativity in prime rings,”
*Bulletin of the Korean Mathematical Society*, vol. 43, no. 1, pp. 101–106, 2006. View at: Publisher Site | Google Scholar - H. Marubayashi, M. Ashraf, N. Rehman, and S. Ali, “On generalized $(\alpha ,\beta )$-derivations in prime rings,”
*Algebra Colloquium*, vol. 17, no. 1, pp. 865–874, 2010. View at: Google Scholar - N. U. Rehman, R. M. AL-Omary, and C. Haetinger, “On Lie structure of prime rings with generalized $(\alpha ,\beta )$-derivations,”
*Boletim da Sociedade Paranaense de Matemática*, vol. 27, no. 2, pp. 43–52, 2009. View at: Google Scholar - M. N. Daif and H. E. Bell, “Remarks on derivations on semiprime rings,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 15, no. 1, pp. 205–206, 1992. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2012 Basudeb Dhara and Atanu Pattanayak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.