Abstract
We introduce the notion of -derivation as a new derivation of -algebra. For an endomorphism map of any -algebra we show that at least one -derivation of exists. Moreover, for such a map, we show that a self-map of is -derivation of if is an associative medial -algebra. For a medial -algebra , is -derivation of if is an outside -derivation of . Finally, we show that if is the identity endomorphism of then the composition of two -derivations of is a -derivation. Moreover, we give a condition to get a commutative composition.
1. Introduction
Derivation is an important area of research in the theory of algebraic structure in mathematics. The theory of derivations of algebraic structures came from the development of Galois theory and the theory of invariants. Many researches have been done on derivations on different algebras (see [1–4]).
Several authors [5–9] have studied derivations in -algebra after the work done in 2004 by Jun and Xin where the notion of derivation in ring and near-ring theory was applied to -algebra [4]. As in [5], for a self-map , , is a left-right derivation (briefly -derivation) of if it satisfies the identity for all If satisfies the identity for all , then is a right-left derivation (briefly -derivation) of If is both - and -derivation, then is a derivation of
Recently, in 2013, a new derivation named -derivation of -algebras was introduced. That is, in general, for any self-map of an algebra , -derivation of is defined by for all and . The map is called an outside -derivation of if it satisfies If the map satisfies the identity , then the map is called an inside -derivation of . If is both outside and inside -derivation of , then it is a -derivation of ([10]).
The notion of -algebra was introduced in [11]. The aim of the paper is to complete the studies on -algebra; in particular, we aim to apply the notion of -derivation on -algebra and obtain some related properties. We start with definitions and propositions on -algebra taken from [11]. Then, we redefine the notion of -derivation in -algebra and prove that every self-map of an associative, medial -algebra is -derivation, where is an endomorphism of . We also show that every self-map of an associative, medial -algebra is -derivation. Then, we show that if is the identity endomorphism of , then, for a medial -algebra, is a -derivation of if is an outside -derivation of . Further, we show that if is the identity endomorphism of and , are both outside (resp., inside) -derivations of , then the composition is an outside (resp., inside) -derivation of and consequently -derivation. We conclude the section with a condition given on two -derivations of to get a commutative composition.
Definition 1. A -algebra is a nonempty set with a constant 0 and a binary operation satisfying the axioms:(1),(2), for all , in .
Proposition 2. If is a -algebra, then the following conditions hold:(1),(2), for any
Proposition 3. Let be a -algebra. Then, the following conditions hold for any : (1),(2),(3)
Definition 4. A -algebra satisfying , for any and , is called a medial -algebra.
Lemma 5. If is a medial -algebra, then, for any , the following axiom holds:
Theorem 6. A -algebra X is medial if and only if it satisfies the following conditions:(1) for all ,(2) for all
2. Results
In this section we will introduce a new derivation of -algebra motivated by [10, Definition 3.1]. We start by defining an endomorphism of -algebra .
Definition 7. Let be a -algebra and let be a self-map of . One says that is an endomorphism if Throughout the paper, is a self-map of -algebra defined by for all , and is an endomorphism self-map of unless otherwise mentioned.
For elements and of a -algebra , denote by By considering that in -algebra, we redefine the notion of -derivation in [10] to get the following definition.
Definition 8. A map is called an outside -derivation of ifIf the map satisfies the following identity:then the map is called an inside -derivation of . If is both an outside and inside -derivation of , then is a -derivation of .
Remark 9. If is -derivation, then
Example 10. Consider the -algebra given by Cayley table (Table 1).
Define an endomorphism:If , then Table 2 shows that is an outside and an inside -derivation of Hence, is a -derivation of
If we take , then is not an outside -derivation of or an inside -derivation of since and
Example 11. Let . Consider the -algebra given by Cayley table (Table 3).
Define an endomorphism:It can be shown by direct calculation that is -derivation of for all
Proposition 12. For any -algebra , there exists at least one -derivation of , that is, the map
Proof. Let ; and We also have Hence, is -derivation of
Proposition 13. If is an associative -algebra, then is an outside -derivation of , for all
Proof. We have and , as is associative. Hence, is an outside -derivation of .
Proposition 14. If is a medial -algebra, then is an inside -derivation of , for all
Proof. Since and , as is medial, therefore, is an inside -derivation of .
The next theorem follows from Propositions 13 and 14.
Theorem 15. Let be an associative medial -algebra; then is a -derivation of , for all
Next we provide an alternative proof of Theorem 15.
Theorem 16. Let be an associative medial -algebra. Then, is both an outside -derivation of and an inside -derivation of for any .
Proof. Let . Then, on one hand, we have By Definition 8 is an outside -derivation of .
On the other hand, Therefore, is an inside -derivation of
Using Proposition 3, we get the following.
Proposition 17. If is an outside (resp., inside) -derivation of , then (resp., .
Proof. It is obvious.
Theorem 18. Let be a medial -algebra. If is an outside -derivation of , then is a -derivation of .
Proof. From Proposition 14, we know that is an inside -derivation of . Thus, is a -derivation of .
Definition 19. A map is said to be regular if
Proposition 20. Let be a -derivation of If either or , then is a regular derivation.
Proof. Since is a -derivation, we have Consider that ; then Similarly, if , we have This proves that is a regular derivation.
Proposition 21. Let be a regular -derivation of ; then
Proof. Since is a regular -derivation of , then So Therefore, from Proposition 3.
Definition 22. Let be a -algebra and let be two self-maps of . Define by
Proposition 23. Let be a -algebra and let be the identity endomorphism of . If are outside -derivations of , then is also an outside -derivation.
Proof. Consider the element . Then . As are outside -derivations, we have Thus, is an outside -derivation.
Similarly, we can prove the following proposition.
Proposition 24. For a -algebra , let be the identity endomorphism of . If are inside -derivations of , then is also an inside -derivation.
Combining Proposition 23 and Proposition 24 we have the following theorem.
Theorem 25. Let be a -algebra and let be the identity endomorphism of . If are both outside (resp., inside) -derivations of , then the composition is an outside (resp., inside) -derivation of .
Proposition 26. Let be a -algebra and let be -derivations of such that and ; then .
Proof. Consider as an outside -derivation of and as an inside -derivation of ; then for all we haveOn the other hand,From (11) and (12), we can see that By putting , we get Hence,
3. Conclusion
In this paper, the notion of -derivation of -algebra is introduced and some related properties are investigated. The main results are Theorems 15 and 18 where we show that a self-map of a -algebra is -derivation if -algebra satisfies some properties. In Theorem 25, we show that in -algebra the composition of two -derivations of is -derivation if is identity endomorphism of Moreover, we give in Proposition 26 a condition on two -derivations of to get a commutative composition.
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.