On multiplicative centrally-extended maps on semi-prime rings

ABSTRACT In this paper, we show that for semi-prime rings of two-torsion free and 6-centrally torsion free, given a multiplicative centrally-extended derivation δ and a multiplicative centrally-extended epimorphism ϕ we can find a central ideal K and maps such as Δ and Φ are multiplicative derivation and multiplicative epimorphism, respectively.


Introduction
Let S be a ring. A mapping σ : S → S is a derivation if it satisfies σ (r + s) = σ (r) + σ (s) and σ (rs) = σ (r)s + rσ (s) for all r, s ∈ S. During the previous few decades, many researchers have dealt with the idea of studying rings, especially the prime and semi-prime kinds endowed with derivations and other related kinds of maps like generalized derivations, α−derivations, left multipliers, and many other types.
Recently in Ref. [1], Bell and Daif gave two concepts. Firstly, a centrally-extended derivation (CE-derivation) of a ring S with centre ξ(S) to be a mapping of S satisfying for each r, s ∈ S, (r + s) − (r) − (s) ∈ ξ(S) and (rs) − (r)s − r (s) ∈ ξ(S). Secondly, a centrally-extended endomorphism (CE-endomorphism) of a ring S with centre ξ(S) to be a mapping of S satisfying for each r, s ∈ S, (r + s) − (r) − (s) ∈ ξ(S) and (rs) − (r) (s) ∈ ξ(S). They discussed the existence of such maps, which are a proper generalization of derivations and endomorphisms. Moreover, they studied their influence on ξ(S) and ring commutativity.
As straightforward generalizations of Ref. [1], the authors in Refs. [2,3] gave the definitions of CE-reverse derivations, CE-generalized reverse derivations, and CEgeneralized derivations endowed with involution. They got similar results as Bell and Daif did in Ref. [1]. In Refs. [4,5], the authors generalize well-known results related to derivations using the new mapping given in Ref. [1].
In Ref. [6], Martindale gave the concept of multiplicative isomorphism as a one-to-one mapping ϑ of a ring R onto a ring S such that ϑ(rt) = ϑ(r)ϑ(t) for all r, t ∈ R is called a multiplicative isomorphism of R onto S. He answered his question: "When is a multiplicative mapping additive?" for a ring S which satisfies some conditions. In Ref. [7], Daif answered that question when the mapping is a multiplicative derivation on S. Generalizations of this idea have been done for the cases of multiplicative-generalized derivations and multiplicative left centralizers in Refs. [8,9].
In Ref. [10], the authors introduced the notion of a multiplicative CE-derivation to be a mapping δ defined on a ring S such as δ(rt) − δ(r)t − rδ(t) ∈ ξ(S) for all r, t ∈ S, i.e. δ(rt) = δ(r)t + rδ(t) + π δ (r, t) where π δ (r, t) is a central element depends on the choice of r and t that are related to the map δ. They asked the following natural question on a multiplicative CE-derivation: "When is a multiplicative CE-derivation a CE-derivation?" Under some conditions, they gave an affirmative answer to this question. As a parallel idea to that given in Ref. [10], we introduce the following definition: is a central element depends on the choice of r and t that are related to the map φ.
In this paper, we show, under some conditions, that every multiplicative CE-derivation (multiplicative CEepimorphism) is a multiplicative derivation (multiplicative epimorphism) in the sense of Ref. [7]. Furthermore, we indicate that S contains ideals of certain types.
The following example shows that multiplicative CE-derivations are a proper generalization of CEderivations.

Example 1.1: Let D be any integral domain,
S is a ring with centre Define the following maps δ, φ : S → S given by It is easily to show that δ is a multiplicative CE-derivation but not a CE-additive map and so it is not a CE-derivation and φ is a multiplicative CE-endomorphism but not a CE-additive map and so it is not a CE-endomorphism. Also, δ(ξ(S)), φ(ξ(S)) ⊆ ξ(S).

Remark 1.1:
An important consideration would be ideals in the centre of S. If I is an ideal in ξ(S), then any map f : S → I is a multiplicative CE-derivation. Furthermore, δ + f , where δ is any multiplicative CE-derivation, will also be a multiplicative CE-derivation. Thus it seems that one cannot hope to say anything specific about the image of f. For example, f (0) does not need to be zero. It can be any element in an ideal in the centre of S. One way to hope to get specific results is to assume that S has no non-zero central ideals.
Let S be a two-torsion free, 6-centrally torsion-free semi-prime ring with a multiplicative CE-derivation δ (a multiplicative CE-epimorphism φ). We show that is a multiplicative derivation mod K ( is a multiplicative epimorphism mod K).
For notation, we use A to indicate the ideal created by a set A. In our work, we suppose that S is a semi-prime ring and φ : S → S is a multiplicative CE-endomorphism. We use the congruence r ≡ s to mean r − s ∈ ξ(S). Also, r ≡ s doesn't imply rz ≡ sz or zr ≡ zs where z ∈ ξ(S).

Preliminaries
We begin this section with some preliminaries which play a substantial role in the main theorem's proof.

Remark 2.1:
The centre of a semi-prime ring has no non-zero nilpotent elements.

Remark 2.2:
In Lemma 2.1, one can replace m by φ(m) and n by φ(n), where φ is a multiplicative CEendomorphism defined in Definition 1.1. Therefore, Thus,
Proof: Expanding δ(mnl) in the two possible associations gives: Subtracting (1) from (2) gives: Since π δ (S, S) is contained in the centre of S. We have mπ δ (n, l) ≡ lπ δ (m, n). The other congruence is immediate from this one.
In Lemma 2.2, replacing the multiplicative CEderivation δ by the multiplicative CE-endomorphism φ and using an analogous proof, we get Proof: Expanding δ(rstu) in two different associations gives: Subtracting (4) from (3) gives: In Lemma 2.4, replacing δ by φ and using Lemma 2.3, one can get an analogous proof for the following lemma: Then h(r, s, t, u) ≡ 0.
f (t, u, r, s) = tuπ δ (r, s) − urπ δ (s, t) Adding (6) to (7) gives: Looking at the map g(r, s, t, u) = [r, s]π δ (t, u), we know that g(r, s, t, u) ≡ −g(s, t, u, r). It is well known also that g(r, s, t, u) ≡ −g(s, r, t, u). Since the two permutations (12) and (1234) generate all members of the symmetric group on four elements, we have shown that g(r, s, t, u) is an alternating map on its four arguments modulo the centre of S.
The following lemma arises from the definition of h in Lemma 2.5 with a similar proof of Lemma 2.6.

Lemma 2.7:
For all r, s, t, u ∈ S, [φ(r), φ(s)]π φ (t, u) is an alternating map on its four arguments modulo the centre of S. Definition 2.1: A ring S is said to be n-centrally torsionfree if nr ≡ 0 mod ξ(S) gives r ≡ 0 for any r ∈ S.
Again, using the definition of h in Lemmas 2.5, 2.7, and a similar proof to the proof of Lemma 2.8 gives:

Lemma 2.10: Let S be a two-torsion-free and a 6centrally torsion-free semi-prime ring. For all r, s, t, u ∈ S,
Throughout the rest of this paper, the map φ will always refer to a multiplicative CE-epimorphism.

S(Sk) ⊆ (SS)k ⊆ ξ(S).
Then K is an ideal in S.
(2) For k ∈ K and x ∈ S, by the definition of the set K, Lemma 2.15: Let S be a two-torsion-free and a 6centrally torsion-free semi-prime ring. Then the ideal K, defined in Lemma 2.14, satisfies the following.

Main results
It is well known that ξ(S) is preserved by derivations and epimorphisms. In Ref. [1], the invariance problem for ξ(S) by CE-derivations and CE-epimorphisms was studied. The authors showed that CE-derivation and CEepimorphism do not preserve the centre of the ring in general (see [1,Examples 3.1 and 3.2]). They showed that if a ring does not contain non-zero central nilpotent elements, then every CE-derivation and every CEepimorphism preserves the centre (see [1,Theorems 3.3 and 3.4]). Also, they proved that every CE-derivation and every CE-epimorphism that preserves the centre preserves the set K. Similarly, we will show, in the following theorem, that the maps δ and φ preserve ξ(S) as well as the ideal K. Proof: (1) For z ∈ ξ(S) and ∈ S, we have and Subtracting (17)  (2) Let k ∈ K and a, b ∈ S. Then we have and Adding (18) and (19) and using part (1) and the definition of K, we get δ(k)a + bδ(k) ∈ ξ(S). Therefore, δ(K) ⊆ K. Again, similar arguments to the previous case, give us φ(K) ⊆ K.
The following example shows that the conditions two-torsion-free and a 6-centrally torsion-free semiprime are necessarily in the previous theorem. Clearly, is a multiplicative CE-derivation (multiplicative CE-endomorphism) and a CE-additive map. A CEderivation (CE-endomorphism). Also, both ξ(S) and K are not invariant under .
Using (2) of Lemma 2.15, one can easily show that the maps , : S/K → S/K that are given by (s + K) = δ(s) + K, and (s + K) = φ(s) + K are well-defined. Also, π δ (S, S) and π φ (S, S) ⊆ K, by (1) of Lemma 2.15. So, we arrive at the proof of the following main result.  Now, let S contains no non-zero central ideals, δ be a CE-derivation, and φ be a CE-epimorphism. Then δ is additive by Bell and Daif [1,Theorem 2.4], and φ is additive by Bell and Daif [1,Theorem 2.7]. Also, the ideal K will be the zero ideal. So, the following result comes immediately. Corollary 3.1: Let S be a semi-prime ring that has no non-zero central ideals. Then every CE-derivation δ (CEepimorphism φ) is a derivation (an epimorphism).

Conclusion
Let S be a two-torsion-free and a 6-centrally torsionfree semi-prime ring, and δ, φ are multiplicative CEderivation and multiplicative CE-epimorphism, respectively. Then the set K = {r ∈ S | rS + Sr ⊆ ξ(S)} is a central ideal and the following results are satisfied (i) δ and φ preserve the centre ξ(S), (ii) δ and φ preserve the ideal K, (iii) There exists a multiplicative derivation : S/K → S/K given by (s + K) = δ(s) + K and a multiplicative epimorphism : S/K → S/K given by (s + K) = φ(s) + K.