Abstract
Let \(R\) will be a \(2\)-torsion free \(*\)-prime ring and \(\alpha ,\beta \in AutR.\) \(F\) be a nonzero generalized \((\alpha ,\beta )\)-derivation of \(R\) with associated nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) and \(J\) be a nonzero \(*\)-Jordan ideal and a subring of \(R.\) In the present paper, we shall prove that \(R\) is commutative if any one of the following holds: (i)\([F(u),u]_{\alpha ,\beta }=0,\) (ii)\(F(u)\alpha (u)=\beta (u)d(u),\) (iii)\(F(u^{2})=\pm \alpha (u^{2}),\) (iv)\(F(u^{2})=2d(u)\alpha (u),\) (v)\(d(u^{2})=2F(u)\alpha (u),\) for all \(u\in U.\)
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1 Introduction
Let \(R\) will be an associative ring with center \(Z.\) For any \(x,y\in R\) the symbol \([x,y]\) represents commutator \(xy-yx\) and the Jordan product \(xoy=xy+yx.\) Recall that a ring \(R\) is prime if \(xRy=0\) implies \(x=0\) or \(y=0.\) An additive mapping \(*:R\rightarrow R\) is called an involution if \((xy)^{*}=y^{*}x^{*}\) and \((x^{*})^{*}=x\) for all \(x,y\in R.\) A ring equipped with an involution is called a ring with involution or \(*\)-ring. A ring with an involution is said to \(*\)-prime if \(xRy=xRy^{*}=0\) or \(xRy=x^{*}Ry=0\) implies that \(x=0\) or \(y=0.\) Every prime ring with an involution is \(*\)-prime but the converse need not hold general. An example due to Oukhtite [7] justifies the above statement that is, \(R\) be a prime ring, \(S=R\times R^{o}\) where \(R^{o}\) is the opposite ring of \(R.\) Define involution \(*\) on \(S\) as \(*(x,y)=(y,x).\) \(S\) is \(*\)-prime, but not prime. This example shows that \(*\)-prime rings constitute a more general class of prime rings. In all that follows the symbol \(S_{a_{*}}(R),\) first introduced by Oukhtite, will denote the set of symmetric and skew symmetric elements of \(R,\) i.e. \(S_{a_{*}}(R)=\{x\in R\mid x^{*}=\pm x\}.\)
An additive subgroup \(J\) of \(R\) is said to be a Jordan ideal of \(R\) if \(uor\in J,\) for all \(u\in J,r\in R.\) A Jordan ideal is said to be a \(*\)-Jordan ideal if \(J^{*}=J.\) Let \(I\) be the ring of integers. Set
and
We define the following maps on \(R:\) \(*\left( \small \begin{array}{ll} a &{} 0\\ c &{} d \end{array} \right) =\left( \small \begin{array} [c]{ll} d &{} 0\\ -c &{} a \end{array} \right) \). Then it is easy to see that \(J\) is a nonzero \(*\)-Jordan ideal of \(R.\)
An additive mapping \(d:R\rightarrow R\) is called a derivation if \(d(xy)=d(x)y+xd(y)\) holds for all \(x,y\in R.\) For a fixed \(a\in R,\) the mapping \(I_{a}:R\rightarrow R\) given by \(I_{a}(x)=[a,x]\) is a derivation which is said to be an inner derivation. An additive mapping \(f:R\rightarrow R\) is called a generalized derivation if there exists a derivation \(d:R\rightarrow R\) such that
This definition was given by Bresar in [3]. Let \(\alpha \) and \(\beta \) be any two automorphisms of \(R.\) An additive mapping \(d:R\rightarrow R\) is called a \((\alpha ,\beta )\)-derivation if \(d(xy)=d(x)\alpha (y)+\beta (x)d(y)\) holds for all \(x,y\in R.\) Inspired by the definition \((\alpha ,\beta )\)-derivation, the notion of generalized derivation was extended as follows: Let \(\alpha \) and \(\beta \) be any two automorphisms of \(R.\) An additive mapping \(f:R\rightarrow R\) is called a generalized \((\alpha ,\beta )\)-derivation on \(R\) if there exists a \((\alpha ,\beta )\)-derivation \(d:R\rightarrow R\) such that
Of course a generalized (1,1)-derivation is a generalized derivation on \(R,\) where 1 is the identity mapping on R.
Let \(S\) be a nonempty subset of \(R\). A mapping \(F\) from \(R\) to \(R\) is called centralizing on \(S\) if \([F(x),x]\in Z,\) for all \(x\in S\) and is called commuting on \(S\) if \([F(x),x]=0,\) for all \(x\in S.\) The study of centralizing and commuting mappings was initiated by Posner in [12] (Posner’s second theorem). Several authors have proved commutativity theorems for prime rings or semiprime rings admitting automorphisms or derivations which are centralizing and commuting on appropriate subsets of \(R\) (see, e.g., [1, 2, 4, 6] and references therein). Recently, Oukhtite et al. [8, Theorem 1] proved Posner’s second theorem to rings with involution in the case of characteristic not 2: Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(U\) a square closed \(*\)-Lie ideal of \(R.\) If \(R\) admits a nonzero derivation \(d\) centralizing on \(U,\) then \(U\subseteq Z\). They generalized this theorem to generalized derivations centralizing on Jordan ideals in rings with involution [11, Theorem 1]. Also, in [5], the authors investigated some commutativity theorems for Jordan ideals in rings with generalized derivation. In the present paper, we shall study these results Jordan ideals of \(*\)-prime rings with generalized \(( \alpha ,\beta )\)-derivation of \(R.\)
2 Preliminaries
Throughout the paper, \(R\) will be a 2-torsion free \(*\)-prime ring and \(\alpha ,\beta \in AutR.\) \(F\) be a nonzero generalized \(( \alpha ,\beta )\)-derivation of \(R\) with associated nonzero \(( \alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) and \(J\) be a nonzero \(*\)-Jordan ideal and a subring of \(R.\) Also, we will make some extensive use of the basic commutator identities:
Lemma 1
[9, Lemma 2] Let \(R\) be a \(2\)-torsion free \(*\)-prime ring, \(J\) a nonzero \(*\)-Jordan ideal of \(R\) and \(a,b\in R.\) If \(aJb=a^{*}Jb=0\), then \(a=0\) or \(b=0\).
Lemma 2
[9, Lemma 3] Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal of \(R.\) If \([J,J]=0,\) then \(J\subseteq Z.\)
Lemma 3
[10, Lemma 3] Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal of \(R.\) If \(J\subseteq Z,\) then \(R\) is commutative.
Lemma 4
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring, \(J\) a nonzero \(*\)-Jordan ideal of \(R\) and \(d\) a nonzero \(( \alpha ,\beta ) \)-derivation of \(R.\) If \(d\) commutes with \(*\) and \(d(J)=0,\) then \(R\) is commutative.
Proof
By the hypothesis, we obtain that
Expanding this term and using the hypothesis, we get
Replacing \(r\) by \(2rv,v\in J\) in (2.1) and using (2.1), \(d(J)=0,\) we have
Substituting \(rs,s\in R\) for \(r\) in this equation and using this, we find that \(d(r)\alpha (s)\alpha ([v,u])=0,\) and so
Writing \(r^{*}\) by \(r\) in the last equation, we get
Since \(d\) commutes with \(*\), the last equation follows
Appliying the \(*\)-primenes of \(R,\) because of (2.2) and (2.3), we conclude that \(d(r)=0\) or \([v,u])=0,\) for all \(u,v\in J,r\in R.\) Since \(d\) is a nonzero \(( \alpha ,\beta )\)-derivation of \(R,\) we arrive at \([J,J]=0,\) and so \(R\) is commutative by Lemmas 2 and 3. \(\square \)
3 Results
Theorem 1
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero generalized \((\alpha ,\beta )\)-derivation \(F\) associated with nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \([F(u),u]_{\alpha ,\beta }=0,\) for all \(u\in J,\) then \(R\) is commutative.
Proof
Suppose that
Linearizing (3.1) and using this, we obtain that
Replacing \(v\) by \(vu\) in (3.1), we get
That is
Now combining (3.1) and (3.2) in the last equation, we find that
Again replace \(v\) by \(vw\) in (3.3) and use (3.3), to get
Since \(\beta \) is an automorphism of \(R,\) we see that
Since \(J\) is a nonzero \(*\)-Jordan ideal of \(R\) yields that
Hence we get
By Lemma 1, we get either \([ v,u] =0,\) for all \(v\in J\) or \(d( u) =0\) for each \(u\in J\cap S_{a_{*}}(R).\) Let \(u\in J,\) as \(u+u^{*},u-u^{*}\in J\cap S_{a_{*}}(R)\) and \([ v,u\pm u^{*}] =0,\) for all \(v\in J\) or \(d(u\pm u^{*})=0.\) Hence we have \([ v,u] =0\) or \(d(u)=0,\) for all \(u,v\in J.\) We obtain that \(J\) is union of two additive subgroups of \(U\) such that
and
Morever, \(J\) is the set-theoretic union of \(K\) and \(L.\) But a group can not be the set-theoretic union of two proper subgroups, hence \(K=J\) or \(L=J.\) In the former case, we get \(R\) is commutative by Lemma 4. In the latter case, \([ J,J] =( 0).\) That is \(J\subseteq Z\) by Lemma 2 and \(R\) is commutative by Lemma 3. This completes the proof. \(\square \)
In the view of Theorem 1, we get the following result:
Corollary 1
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \([d(u),u]_{\alpha ,\beta }=0,\) for all \(u\in J,\) then \(R\) is commutative.
Theorem 2
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero generalized \((\alpha ,\beta )\)-derivation \(F\) associated with nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \(F(u)\alpha (u)=\beta (u)d(u),\) for all \(u\in J,\) then \(R\) is commutative.
Proof
We have
Replacing \(u\) by \(u+v\) in (3.5) and using this, we get
Writting \(vu\) for \(v\) in (3.6) and using (3.6), we obtain that
Taking \(wv\) instead of \(v\) in the above equation and using this, we have
Hence we arrive at
Since \(J\) is a nonzero \(*\)-Jordan ideal of \(R\) yields that
Therefore, we get
The similar arguments as used after equation (3.4), we get the required result. \(\square \)
Theorem 3
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero generalized \((\alpha ,\beta )\)-derivation \(F\) associated with nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \(F(u^{2})=\pm \alpha (u^{2}),\) for all \(u\in J,\) then \(R\) is commutative.
Proof
Linearizing the hypothesis, we get
Replacing \(v\) by \(vu,\) \(u\in J\) in (3.7) and appliying this equation, we arrive at
Writting \(vw\) for \(v\) in (3.8) and using (3.8), we obtain that
and so
Since \(J\) is a nonzero \(*\)-Jordan ideal of \(R\) yields that
Therefore, we get
The similar arguments as used after equation (3.4), we get the required result. \(\square \)
Theorem 4
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero generalized \((\alpha ,\beta )\)-derivation \(F\) associated with nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \(F(u^{2})=2d(u)\alpha (u),\) for all \(u\in J,\) then \(R\) is commutative.
Proof
We get
That is
Linearizing (3.9) and using this, we obtain
Taking \(vu\) instead of \(v\) in (3.10) and using this equation, we have
Letting \(v\) by \(wv\) in (3.11) and using (3.11), we arrive at
That is
Since \(J\) is a nonzero \(*\)-Jordan ideal of \(R\) yields that
Therefore, we get
Further application of similar arguments as used after (3.4), we get the required result. \(\square \)
Theorem 5
Let \(R\) be a \(2\)-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero generalized \((\alpha ,\beta )\)-derivation \(F\) associated with nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \(d(u^{2})=2F(u)\alpha (u),\) for all \(u\in J,\) then \(R\) is commutative.
Proof
Expanding \(d(u^{2})=2F(u)\alpha (u),\) we have
Linearizing (3.13) and using this, we obtain
Writing \(vu\) instead of \(v\) in (3.14) and using this equation, we have
Replacing \(v\) by \(wv\) in the last equation and using this, we arrive at
The similar arguments as used after equation (3.12) in the proof of Theorem 4, we get the required result. \(\square \)
We can give the following corollary by Theorem 4 (or 5):
Corollary 2.
Let \(R\) be a 2-torsion free \(*\)-prime ring and \(J\) a nonzero \(*\)-Jordan ideal and a subring of \(R.\) If \(R\) admits a nonzero \((\alpha ,\beta )\)-derivation \(d\) which commutes with \(*\) such that \(d(u^{2})=2d(u)\alpha (u),\) for all \(u\in J,\) then \(R\) is commutative.
References
Bell, H.E., Martindale, W.S.: Centralizing mappings of semiprime rings. Can. Math. Bull. 30, 92–101 (1987)
Bresar, M.: Centralizing mappings and derivations in prime rings. J. Algebra 156, 385–394 (1993)
Bresar, M.: On the distance of the compositions of two derivations to the generalized derivations. Glasg. Math. J. 33(1), 89–93 (1991)
Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings. J. Algebra 71, 259–267 (1981)
El-Soufi, M., Serag, A.: Generalized derivations on Jordan ideals in prime rings. Turkish J. Math. (2013, to appear)
Mayne, J.H.: Centralizing automorphisms of prime rings. Can. Math. Bull. 19, 113–115 (1976)
Oukhtite, L., Salhi, S.: On generalized derivations of \(\ast \)-prime rings. Afr. Diaspora J. Math. 5(1), 19–23 (2006)
Oukhtite, L., Salhi, S., Taoufiq, L.: Commutativity conditions on derivations and Lie ideals in \(\sigma \)-prime rings. Beitrage Algebra Geom. 51(1), 275–282 (2010)
Oukhtite, L.: On Jordan ideals and derivations in rings with involution. Comment. Math. Univ. Carol. 51(13), 389–395 (2010)
Oukhtite, L.: Posner’s second theorem for Jordan ideals in rings with involution. Expo. Math. 29(4), 415–419 (2011)
Oukhtite, L., Mamouni, A.: Generalized derivations centralizing on Jordan ideals of rings with involution. Turkish J. Math. (2013, to appear)
Posner, E.C.: Derivations in prime rings. Proc Am. Math. Soc. 8, 1093–1100 (1957)
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Gölbaşi, Ö., Kizilgöz, Ö. Generalized \((\alpha ,\beta )\)-derivations on Jordan ideals in \(*\)-prime rings. Rend. Circ. Mat. Palermo 63, 11–17 (2014). https://doi.org/10.1007/s12215-013-0138-2
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DOI: https://doi.org/10.1007/s12215-013-0138-2