Abstract
Let be a near ring. An additive mapping is said to be a generalized semiderivation on if there exists a semiderivation associated with a function such that and for all . In this paper we prove that prime near rings satisfying identities involving semiderivations are commutative rings, thereby extending some known results on derivations, semiderivations, and generalized derivations. We also prove that there exist no nontrivial generalized semiderivations which act as a homomorphism or as an antihomomorphism on a 3-prime near ring .
1. Introduction
Throughout the paper, denotes a zero-symmetric left near ring with multiplicative centre ; and for any pair of elements , [] denotes the commutator while the symbol denotes the additive commutator . An element of is said to be distributive if , for all . A near ring is called zero-symmetric if , for all (recall that left distributivity yields that ). The near ring is said to be 3-prime if for implies that or . A near ring is called 2-torsion free if has no element of order 2. An additive mapping is said to be a right (resp., left) generalized derivation with associated derivation if (resp., ), for all , and is said to be a generalized derivation with associated derivation on if it is both a right generalized derivation and a left generalized derivation on with associated derivation . Motivated by a definition given by Bergen [1] for rings, we define an additive mapping to be a semiderivation on a near ring if there exists a function such that (i) and (ii) , for all . In case is the identity map on , is of course just a derivation on , so the notion of semiderivation generalizes that of derivation. But the generalization is not trivial; for example, take , where is a zero-symmetric near ring and is a ring. Then the map defined by is a semiderivation associated with function such that . However is not a derivation on . An additive mapping is said to be a generalized semiderivation of if there exists a semiderivation associated with a map such that (i) and (ii) for all . All semiderivations are generalized semiderivations. Moreover, if is the identity map on , then all generalized semiderivations are merely generalized derivations; again the notion of generalized semiderivation generalizes that of generalized derivation. Moreover, the generalization is not trivial as the following example shows.
Example 1. Let be a 2-torsion free left near ring and let Define maps by It can be verified that is a left near ring and is a generalized semiderivation with associated semiderivation and a map associated with . However is not a generalized derivation on .
2. Preliminary Results
We begin with the following Lemmas which are extensively used to prove our main theorems. Unless it is stated otherwise, it will be assumed that is a zero-symmetric 3-prime near ring.
Lemma 2 (see [2, Lemma ]). Let be a 3-prime near ring. (i)If and , then (ii)If then is not a zero divisor.
Lemma 3 (see [2, Lemma ]). If is a 3-prime near ring and contains a nonzero left semigroup ideal, then is a commutative ring.
Lemma 4 (see [3, Theorem ]). Let be a 2-torsion free 3-prime near ring with a nonzero semiderivation associated with a map . If , then is a commutative ring.
Lemma 5. Let be a 3-prime near ring admitting a generalized semiderivation associated with a semiderivation . If is the map associated with such that for all , then satisfies the following partial distributive laws: (i) for all (ii) for all
Proof. (i) Let , and by defining we have On the other hand, Combining both expressions of , we obtain (ii) With a simple calculation of , we obtain the required result.
Lemma 6. Let be a 2-torsion free zero-symmetric 3-prime near ring. If is a nonzero semiderivation of associated with a map which is onto, then .
Proof. Suppose . Then for , we may writeNote that and is onto; we get Since is 2-torsion free, we get Replacing by in the above relation, we get This implies that Thus we obtain that , a contradiction.
The following lemma extends results of Herstein [4, Theorem ] and Bell and Mason [2, Theorem ].
Lemma 7. Let be a 2-torsion free 3-prime near ring admitting a nonzero semiderivation and a map associated with such that is onto and for all . If for all , then is a commutative ring.
Proof. Suppose that Replacing by in (11) and using Lemma 5(ii), we obtain Substituting for in (12) and using (11), we find that Taking instead of in (13) after using (13), we arrive at which can be rewritten as In the light of the 3-primeness of , (15) implies that But contradicts Lemma 6, so is contained in and is a commutative ring by Lemma 4.
3. The Condition
The theorems that we prove in this section are motivated by the results proved in [2, Theorem ], [5, Theorem ], [6, Theorem and ], and [3, Theorem ].
Theorem 8. Let be a 2-torsion free 3-prime near ring with a generalized semiderivation associated with a nonzero semiderivation and onto map associated with such that for all . If , then is a commutative ring.
Proof. Assume that Taking into account Lemma 5(i), we have for all , , , Since , we have for all , , . Thus for all . Let . Choosing such that and noting that , we have . Since is onto, we have . Hence is a commutative ring by Lemma 3. On the other hand if , then for all Hence for all . Since is onto, we have for all . This implies that Left multiplying by , we arrive at Since is a 3-prime near ring, we get We conclude that is a commutative ring by Lemma 7.
Corollary 9 (see [6, Theorem ]). Let be a 2-torsion free 3-prime near ring. If admits a nonzero generalized derivation such that , then is a commutative ring.
Theorem 10. Let be a 3-prime near ring admitting a generalized semiderivation associated with a nonzero semiderivation and onto map associated with such that for all . If , then is abelian.
Proof. Assume that Then By (23), the last equation yields that Hence, which implies that Putting instead of in (27), we get which can be rewritten as Since is a 3-prime near ring and , we get Replacing and by and , respectively, in (30), we obtain Taking instead of in the last equation and using Lemma 5(ii), we get Thus, Again using the fact that is 3-prime and , we find that for all . Since is onto, is abelian.
Theorem 11. Let be a 2-torsion free 3-prime near ring admitting a nonzero generalized semiderivation associated with a nonzero semiderivation and onto map associated with such that for all . If , then is a commutative ring.
Proof. By the hypothesis Replace by in the above relation, and we get This implies that Using Lemma 5(i), we find that Taking instead of in (37) and using (37) Since is onto, we get This implies that Since is a 3-prime near ring, we have If is contained in , then is a commutative ring by Theorem 8. On the other hand, we see that if , then Thus This implies that Replacing by and using the fact that is onto, we get Since is 2-torsion free, using (44) we get Thus we obtain that , a contradiction which completes the proof.
Corollary 12 (see [6, Theorem ]). Let be a 2-torsion free prime near ring. If admits a generalized derivation associated with a nonzero derivation such that for all , then is a commutative ring.
Theorem 13. Let be a 2-torsion free 3-prime near ring. If is a generalized semiderivation of associated with a nonzero semiderivation and an automorphism associated with , then the following assertions are equivalent:(i) for all (ii) for all (iii) is a commutative ring.
Proof. It is obvious that (iii) implies both (i) and (ii).
Now we prove that . By hypothesis Taking instead of in (47) and noting that , we get Using (47) and noting that by (47), then the last equation yields Since is an automorphism, we get Replacing by in (50) and using (50), we arrive at This implies that 3-primeness of yields that either or . In both the cases is a commutative ring by Lemmas 3 and 4, respectively.
Using the similar techniques as above we can show that .
Corollary 14 (see [7, Theorem ]). Let be a 3-prime near ring. If admits a generalized derivation associated with a nonzero derivation such that for all , then is a commutative ring.
Theorem 15. Let be a 2-torsion free 3-prime near ring. If is a generalized semiderivation of with associated semiderivation and an automorphism associated with , then the following assertions are equivalent:(i) for all (ii) for all (iii) is a commutative ring.
Proof. Obviously, (iii) implies both (i) and (ii).
Now we prove that . By hypothesis Replacing by in (53), we arrive at Using (53) and noting that by (53), we find that Arguing in the similar manner as in Theorem 13, we get the result.
Similarly we can prove that .
Corollary 16 (see [7, Theorem ]). Let be 3-prime near ring. If admits a generalized derivation associated with a nonzero derivation such that for all , then is a commutative ring.
The following example shows that the conditions on the hypothesis of the above theorems are not superfluous.
Example 17. Let be a 2-torsion free left near ring and let Define by It can be checked that is a left near ring and is a generalized semiderivation of associated with a semiderivation and onto map associated with satisfying(i),(ii),(iii),(iv),(v)for all . However, is not a commutative ring.
4. Generalized Semiderivations Acting as a Homomorphism or as an Antihomomorphism
In [8], Bell and Kappe proved that if is a semiprime ring and is a derivation on which is either an endomorphism or an antiendomorphism on , then . Of course, derivations which are not endomorphisms or antiendomorphisms on may behave as such on certain subsets of ; for example, any derivation behaves as the zero endomorphism on the subring consisting of all constants (i.e., the elements for which ). In fact in a semiprime ring , may behave as an endomorphism on a proper ideal of . However as noted in [8], the behaviour of is somewhat restricted in the case of a prime ring. Recently the authors in [9] considered -derivation acting as a homomorphism or an antihomomorphism on a nonzero Lie ideal of a prime ring and concluded that . In this section we establish similar results in the setting of a 3-prime near ring admitting a generalized semiderivation.
Theorem 18. Let be a 3-prime near ring. Suppose that is a generalized semiderivation of associated with a semiderivation and onto map associated with such that for all . If acts as a homomorphism on , then either is identity map or .
Proof. By the hypothesis Replacing by in the above relation, we get This implies that Using Lemma 5(ii), we obtain This implies that Thus Therefore, or for all .
In the later case is an identity map. On the other hand suppose that . Then ; that is, for all Replacing by , , and noting that , we have for all . Therefore, or is an identity map.
Theorem 19. Let be a 2-torsion free 3-prime near ring. Suppose that is a generalized semiderivation of associated with a semiderivation and onto map such that for all . If acts as antihomomorphism on , then or is the identity map on and is a commutative ring.
Proof. By the hypothesis Thus Replacing by in the above relation, we obtain By Lemma 5(ii), we have This implies that Replacing by in the above relation, we get Using (68) in the above relation, we get Since is onto, we have Therefore, either or . Hence in either case acts as a homomorphism by Lemma 4 and Theorem 8 which completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.