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 $\mathrm{\Delta },\mathrm{\Phi }:S/K\to S/K$ such as Δ and Φ are multiplicative derivation and multiplicative epimorphism, respectively.

KEYWORDS:

• Derivations
• CE-derivations
• CE-epimorphism

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 16W25
• 16N60
• 16U80

1. Introduction

Let S be a ring. A mapping $\sigma :S\to S$ is a derivation if it satisfies $\sigma (r+s)=\sigma \left(r\right)+\sigma \left(s\right)$ and $\sigma \left(rs\right)=\sigma \left(r\right)s+r\sigma \left(s\right)$ for all $r,s\in S$. 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, $\alpha -$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 $\xi \left(S\right)$ to be a mapping Ω of S satisfying for each $r,s\in S,$ $\mathrm{\Omega }(r+s)-\mathrm{\Omega }\left(r\right)-\mathrm{\Omega }\left(s\right)\in \xi \left(S\right)$ and $\mathrm{\Omega }\left(rs\right)-\mathrm{\Omega }\left(r\right)s-r\mathrm{\Omega }\left(s\right)\in \xi \left(S\right)$. Secondly, a centrally-extended endomorphism (CE-endomorphism) of a ring S with centre $\xi \left(S\right)$ to be a mapping ℏ of S satisfying for each $r,s\in S,\hslash (r+s)-\hslash \left(r\right)-\hslash \left(s\right)\in \xi \left(S\right)$ and $\hslash \left(rs\right)-\hslash \left(r\right)\hslash \left(s\right)\in \xi \left(S\right)$. They discussed the existence of such maps, which are a proper generalization of derivations and endomorphisms. Moreover, they studied their influence on $\xi \left(S\right)$ 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 CE-generalized 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 $\vartheta \left(rt\right)=\vartheta \left(r\right)\vartheta \left(t\right)$ for all $r,t\in R$ 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 $\delta \left(rt\right)-\delta \left(r\right)t-r\delta \left(t\right)\in \xi \left(S\right)$ for all $r,t\in S$, i.e. $\delta \left(rt\right)=\delta \left(r\right)t+r\delta \left(t\right)+{\pi }_{\delta }(r,t)$ where ${\pi }_{\delta }(r,t)$ 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:

Definition 1.1

A multiplicative CE-endomorphism of a ring S is a mapping ϕ of S such as $\varphi \left(rt\right)=\varphi \left(r\right)\varphi \left(t\right)+{\pi }_{\varphi }(r,t)$ for all $r,t\in S$ where ${\pi }_{\varphi }(r,t)$ 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 CE-epimorphism) 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 CE-derivations.

Example 1.1

Let D be any integral domain, $S=\{\left[\begin{array}{ccc}0& r& s\\ 0& 0& t\\ 0& 0& 0\end{array}\right];r,s,t\in D\}.$ S is a ring with centre $\xi \left(S\right)=\{\left[\begin{array}{ccc}0& 0& u\\ 0& 0& 0\\ 0& 0& 0\end{array}\right];u\in D\}.$ Define the following maps $\delta ,\varphi :S\to S$ given by $\begin{array}{rl}\delta \left(\left[\begin{array}{ccc}0& r& s\\ 0& 0& t\\ 0& 0& 0\end{array}\right]\right)& =\left[\begin{array}{ccc}0& rt& s\\ 0& 0& rt\\ 0& 0& 0\end{array}\right]\mathrm{a}\mathrm{n}\mathrm{d}\\ \varphi \left(\left[\begin{array}{ccc}0& r& s\\ 0& 0& t\\ 0& 0& 0\end{array}\right]\right)& =\left[\begin{array}{ccc}0& rt& rt\\ 0& 0& rt\\ 0& 0& 0\end{array}\right].\end{array}$ 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, $\delta \left(\xi \right(S\left)\right),\varphi \left(\xi \right(S\left)\right)\subseteq \xi \left(S\right)$.

Remark 1.1

An important consideration would be ideals in the centre of S. If I is an ideal in $\xi \left(S\right)$, then any map $f:S\to I$ is a multiplicative CE-derivation. Furthermore, $\delta +f$, 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, $f\left(0\right)$ 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 the set $K=\{r\in S|rS+Sr\subseteq \xi (S\left)\right\}$ is a central ideal. The map $\mathrm{\Delta }(r+K)=\delta \left(r\right)+K$ $\left(\mathrm{\Phi }\right(r+K)=\varphi (r)+K)$ is well-defined on the cosets S/K. Δ is a multiplicative derivation $\mathrm{mod}K$ (Φ is a multiplicative epimorphism $\mathrm{mod}K$).

For notation, we use $〈A〉$ to indicate the ideal created by a set A. Thus $〈[S,S]〉$ is the ideal created by all commutators and $〈\left[\right[S,S],S]〉$ is the ideal generated by all double commutators. We use $\xi \left(S\right)$ to indicate the centre of the ring S. We use the identities $[ax,u]=a[x,u]+[a,u]x$ and $[a,xu]=x[a,u]+[a,x]u$ frequently. In our work, we suppose that S is a semi-prime ring and $\varphi :S\to S$ is a multiplicative CE-endomorphism. We use the congruence $r\equiv s$ to mean $r-s\in \xi \left(S\right)$. Also, $r\equiv s$ doesn't imply $rz\equiv sz\mathrm{o}\mathrm{r}zr\equiv zs$ where $z\in \xi \left(S\right)$.

2. 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.

Lemma 2.1

For all $m,l\in S,$ $2[m,l{]}^2\in 〈[[S,S],S]〉$.

Proof.

Since $\begin{array}{rl}& \left[\right[m^2,l],l]\\ & =\left[m\right[m,l]+[m,l]m,l]=\left[m\right[m,l],l]+\left[\right[m,l]m,l]\\ & =m\left[\right[m,l],l]+[m,l][m,l]+[m,l][m,l]+\left[\right[m,l],l]m\\ & =2[m,l][m,l]+m\left[\right[m,l],l]+\left[\right[m,l],l]m.\end{array}$ Thus, $2[m,l][m,l]=\left[\right[m^2,l],l]-m\left[\right[m,l],l]-\left[\right[m,l],l]m$.

Thus, $2[m,l][m,l]\in 〈\left[\right[S,S],S]〉$.

Remark 2.2

In Lemma 2.1, one can replace m by $\varphi \left(m\right)$ and n by $\varphi \left(n\right),$ where ϕ is a multiplicative CE-endomorphism defined in Definition 1.1. Therefore, $\begin{array}{rl}& 2\left[\varphi \right(m),\varphi (n\left)\right]\left[\varphi \right(m),\varphi (n\left)\right]\\ & =\left[\right[\varphi (m{)}^2,\varphi (n\left)\right],\varphi \left(n\right)]-\varphi (m\left)\right[\left[\varphi \right(m),\varphi (n\left)\right],\varphi \left(n\right)]\\ & -\left[\right[\varphi \left(m\right),\varphi \left(n\right)],\varphi (n\left)\right]\varphi \left(m\right)\text{for all}m,n\in S.\end{array}$ Thus, $2\left[\varphi \right(m),\varphi (n\left)\right]\left[\varphi \right(m),\varphi (n\left)\right]\in 〈\left[\right[\varphi \left(S\right),\varphi \left(S\right)],\varphi (S\left)\right]〉.$

Lemma 2.2

For all $m,n,l\in S$, we have $m{\pi }_{\delta }(n,l)\equiv n{\pi }_{\delta }(l,m)\equiv l{\pi }_{\delta }(m,n)$.

Proof.

Expanding $\delta \left(mnl\right)$ in the two possible associations gives: (1) $\begin{array}{rl}\delta \left(\right(mn\left)l\right)& =\delta \left(m\right)nl+m\delta \left(n\right)l+mn\delta \left(l\right)\\ & +{\pi }_{\delta }(m,n)l+{\pi }_{\delta }(mn,l).\end{array}$(1) (2) $\begin{array}{rl}\delta \left(m\right(nl\left)\right)& =\delta \left(m\right)nl+m\delta \left(n\right)l+mn\delta \left(l\right)\\ & +m{\pi }_{\delta }(n,l)+{\pi }_{\delta }(m,nl).\end{array}$(2) Subtracting (1(1) $\begin{array}{rl}\delta \left(\right(mn\left)l\right)& =\delta \left(m\right)nl+m\delta \left(n\right)l+mn\delta \left(l\right)\\ & +{\pi }_{\delta }(m,n)l+{\pi }_{\delta }(mn,l).\end{array}$(1) ) from (2(2) $\begin{array}{rl}\delta \left(m\right(nl\left)\right)& =\delta \left(m\right)nl+m\delta \left(n\right)l+mn\delta \left(l\right)\\ & +m{\pi }_{\delta }(n,l)+{\pi }_{\delta }(m,nl).\end{array}$(2) ) gives: $0={\pi }_{\delta }(m,n)l+{\pi }_{\delta }(mn,l)-m{\pi }_{\delta }(n,l)-{\pi }_{\delta }(m,nl).$ Since ${\pi }_{\delta }(S,S)$ is contained in the centre of S. We have $m{\pi }_{\delta }(n,l)\equiv l{\pi }_{\delta }(m,n)$. The other congruence is immediate from this one.

In Lemma 2.2, replacing the multiplicative CE-derivation δ by the multiplicative CE-endomorphism ϕ and using an analogous proof, we get

Lemma 2.3

$\varphi \left(m\right){\pi }_{\varphi }(n,l)\equiv \varphi \left(n\right){\pi }_{\varphi }(l,m)\equiv \varphi \left(l\right){\pi }_{\varphi }(m,n)$ for all $m,n,l\in S$.

Lemma 2.4

Assuming that $f(r,s,t,u)=rs{\pi }_{\delta }(t,u)-st{\pi }_{\delta }(u,r)+tu{\pi }_{\delta }(r,s)-ru{\pi }_{\delta }(s,t)$. Then $f(r,s,t,u)\equiv 0$.

Proof.

Expanding $\delta \left(rstu\right)$ in two different associations gives: (3) $\begin{array}{rl}\delta \left(\right(\left(rs\right)t\left)u\right)& =\delta \left(r\right)stu+r\delta \left(s\right)tu+rs\delta \left(t\right)u+rst\delta \left(u\right)\\ & +{\pi }_{\delta }(rs,t)u+{\pi }_{\delta }(r,s)tu+{\pi }_{\delta }(rst,u).\end{array}$(3) (4) $\begin{array}{rl}\delta \left(\right(r\left(st\right)\left)u\right)& =\delta \left(r\right)stu+r\delta \left(s\right)tu+rs\delta \left(t\right)u+rst\delta \left(u\right)\\ & +{\pi }_{\delta }(r,st)u+r{\pi }_{\delta }(s,t)u+{\pi }_{\delta }(rst,u).\end{array}$(4) Subtracting (4(4) $\begin{array}{rl}\delta \left(\right(r\left(st\right)\left)u\right)& =\delta \left(r\right)stu+r\delta \left(s\right)tu+rs\delta \left(t\right)u+rst\delta \left(u\right)\\ & +{\pi }_{\delta }(r,st)u+r{\pi }_{\delta }(s,t)u+{\pi }_{\delta }(rst,u).\end{array}$(4) ) from (3(3) $\begin{array}{rl}\delta \left(\right(\left(rs\right)t\left)u\right)& =\delta \left(r\right)stu+r\delta \left(s\right)tu+rs\delta \left(t\right)u+rst\delta \left(u\right)\\ & +{\pi }_{\delta }(rs,t)u+{\pi }_{\delta }(r,s)tu+{\pi }_{\delta }(rst,u).\end{array}$(3) ) gives: ${\pi }_{\delta }(rs,t)u+{\pi }_{\delta }(r,s)tu-{\pi }_{\delta }(r,st)u-r{\pi }_{\delta }(s,t)u=0.$ Using Lemma 2.2, we get: $rs{\pi }_{\delta }(t,u)-st{\pi }_{\delta }(u,r)+tu{\pi }_{\delta }(r,s)-rd{\pi }_{\delta }(s,t)\equiv 0$.

In Lemma 2.4, replacing δ by ϕ and using Lemma 2.3, one can get an analogous proof for the following lemma:

Lemma 2.5

Assuming that: (5) $\begin{array}{rl}h(r,s,t,u)& =\varphi \left(r\right)\varphi \left(s\right){\pi }_{\varphi }(t,u)-\varphi \left(s\right)\varphi \left(t\right){\pi }_{\varphi }(u,r)\\ & +\varphi \left(t\right)\varphi \left(u\right){\pi }_{\varphi }(r,s)-\varphi \left(r\right)\varphi \left(u\right){\pi }_{\varphi }(s,t).\end{array}$(5) Then $h(r,s,t,u)\equiv 0$.

Lemma 2.6

For all $t,s,r,u\in S$, $[r,s]{\pi }_{\delta }(t,u)$ is an alternating map on its four arguments modulo the centre of S.

Proof.

From the definition of f in Lemma 2.4, we get: (6) $\begin{array}{rl}f(s,t,u,r)& =st{\pi }_{\delta }(u,r)-tu{\pi }_{\delta }(r,s)\\ & +ur{\pi }_{\delta }(s,t)-sr{\pi }_{\delta }(t,u)\equiv 0.\end{array}$(6) (7) $\begin{array}{rl}f(t,u,r,s)& =tu{\pi }_{\delta }(r,s)-ur{\pi }_{\delta }(s,t)\\ & +rs{\pi }_{\delta }(t,u)-ts{\pi }_{\delta }(u,r)\equiv 0.\end{array}$(7) Adding (6(6) $\begin{array}{rl}f(s,t,u,r)& =st{\pi }_{\delta }(u,r)-tu{\pi }_{\delta }(r,s)\\ & +ur{\pi }_{\delta }(s,t)-sr{\pi }_{\delta }(t,u)\equiv 0.\end{array}$(6) ) to (7(7) $\begin{array}{rl}f(t,u,r,s)& =tu{\pi }_{\delta }(r,s)-ur{\pi }_{\delta }(s,t)\\ & +rs{\pi }_{\delta }(t,u)-ts{\pi }_{\delta }(u,r)\equiv 0.\end{array}$(7) ) gives: $[r,s]{\pi }_{\delta }(t,u)+[s,t]{\pi }_{\delta }(u,r)\equiv 0.$ Looking at the map $g(r,s,t,u)=[r,s]{\pi }_{\delta }(t,u),$ we know that $g(r,s,t,u)\equiv -g(s,t,u,r)$. It is well known also that $g(r,s,t,u)\equiv -g(s,r,t,u)$. 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 $r,s,t,u\in S$, $\left[\varphi \right(r),\varphi (s\left)\right]{\pi }_{\varphi }(t,u)$ 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 torsion-free if $nr\equiv 0$ $\mathrm{mod}\xi \left(S\right)$ gives $r\equiv 0$ for any $r\in S$.

Lemma 2.8

Let S be a 6-centrally torsion-free semi-prime ring, then for all $r,s,t,u,v\in S$, ${\pi }_{\delta }(r,s)\left[\right[S,S],S]=0$.

Proof.

Using the definition of f in Lemma 2.4, we have (8) $\begin{array}{rl}-f(r,u,t,s)& =-ru{\pi }_{\delta }(t,s)+ut{\pi }_{\delta }(s,r)\\ & -ts{\pi }_{\delta }(r,u)+rs{\pi }_{\delta }(u,t)\equiv 0.\end{array}$(8) (9) $\begin{array}{rl}+f(u,s,r,t)& =+us{\pi }_{\delta }(r,t)-sr{\pi }_{\delta }(t,u)\\ & +rt{\pi }_{\delta }(u,s)-ut{\pi }_{\delta }(s,r)\equiv 0.\end{array}$(9) (10) $\begin{array}{rl}-f(u,s,t,r)& =-us{\pi }_{\delta }(t,r)+st{\pi }_{\delta }(r,u)\\ & -tr{\pi }_{\delta }(u,s)+ur{\pi }_{\delta }(s,t)\equiv 0.\end{array}$(10) (11) $\begin{array}{rl}-f(u,t,r,s)& =-ut{\pi }_{\delta }(r,s)+tr{\pi }_{\delta }(s,u)\\ & -rs{\pi }_{\delta }(u,t)+us{\pi }_{\delta }(t,r)\equiv 0.\end{array}$(11) (12) $\begin{array}{rl}+f(u,r,t,s)& =+ur{\pi }_{\delta }(t,s)-rt{\pi }_{\delta }(s,u)\\ & +ts{\pi }_{\delta }(u,r)-us{\pi }_{\delta }(r,t)\equiv 0.\end{array}$(12) (13) $\begin{array}{rl}-f(u,r,s,t)& =-ur{\pi }_{\delta }(s,t)+rs{\pi }_{\delta }(t,u)\\ & -st{\pi }_{\delta }(u,r)+ut{\pi }_{\delta }(r,s)\equiv 0.\end{array}$(13) Adding Equations (8(8) $\begin{array}{rl}-f(r,u,t,s)& =-ru{\pi }_{\delta }(t,s)+ut{\pi }_{\delta }(s,r)\\ & -ts{\pi }_{\delta }(r,u)+rs{\pi }_{\delta }(u,t)\equiv 0.\end{array}$(8) )–(13(13) $\begin{array}{rl}-f(u,r,s,t)& =-ur{\pi }_{\delta }(s,t)+rs{\pi }_{\delta }(t,u)\\ & -st{\pi }_{\delta }(u,r)+ut{\pi }_{\delta }(r,s)\equiv 0.\end{array}$(13) ) gives: $\begin{array}{rl}0& \equiv [u,r]{\pi }_{\delta }(t,s)+[s,t]{\pi }_{\delta }(r,u)+[r,s]{\pi }_{\delta }(t,u)\\ & +[r,t]{\pi }_{\delta }(u,s)+[t,r]{\pi }_{\delta }(s,u)+[t,s]{\pi }_{\delta }(u,r).\end{array}$ These are all even permutations of $[r,s]{\pi }_{\delta }(t,u)$. Using Lemma 2.6, we have $6[r,s]{\pi }_{\delta }(t,u)\equiv 0,$ which implies $[r,s]{\pi }_{\delta }(t,u)\equiv 0,$ by assumption, that is ${\pi }_{\delta }(r,s)[t,u]\in \xi \left(S\right)$. Therefore ${\pi }_{\delta }(r,s)\left[\right[t,u],v]=\left[{\pi }_{\delta }\right(r,s\left)\right[t,u],v]=0.\text{■}$

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 2.9

Let S be a 6-centrally torsion-free semi-prime ring. Then for all $r,s,t,u\in S$, ${\pi }_{\varphi }(r,s)\left[\right[\varphi \left(S\right),\varphi \left(S\right)],\varphi (S\left)\right]=0$.

Lemma 2.10

Let S be a two-torsion-free and a 6-centrally torsion-free semi-prime ring. For all $r,s,t,u\in S$, ${\pi }_{\delta }(r,s)[t,u]=0$.

Proof.

By Lemmas 2.1 and 2.8, we get $2\left\{{\pi }_{\delta }\right(r,s\left)\right[t,u]{\}}^2\in {\pi }_{\delta }(r,s)〈[[S,S],S]〉=\{0\}.$ Since S is two-torsion-free, this gives $\left\{{\pi }_{\delta }\right(r,s\left)\right[t,u]{\}}^2=0$. Hence ${\pi }_{\delta }(r,s)[t,u]\in \xi \left(S\right)$ that squares to zero. Using Remark 2.1, we arrive to ${\pi }_{\delta }(r,s)[t,u]=0$.

By Remark 2.2, Lemma 2.9, Remark 2.1, and similar parallel steps to the proof of Lemma 2.10 give:

Lemma 2.11

Let S be a two-torsion-free and a 6-centrally torsion-free semi-prime ring, and ϕ be a multiplicative CE-endomorphism. For all $r,s,t,u\in S$, ${\pi }_{\varphi }(r,s)\left[\varphi \right(t),\varphi (u\left)\right]=0$. Moreover, if ϕ is a multiplicative CE-epimorphism, then ${\pi }_{\varphi }(r,s)[t,u]=0$.

Lemma 2.12

Let ℓ be an element of a semi-prime ring S. If $\ell [S,S]=0$ and $[S,S]\ell =0,$ then $〈\ell 〉\subseteq \xi \left(S\right)$.

Proof.

Assume $\ell [S,S]=[S,S]\ell =0$. Then the following are satisfied for all $a,b\in S$.

1. $[\ell ,a][\ell ,b]=\left[\right[\ell ,a]\ell ,b]-\left[\right[\ell ,a],b]\ell =0$.

2. $〈[\ell ,S]+[\ell ,S]S〉=〈[\ell ,S]+S[\ell ,S]〉$.

The equality holds because $[\ell ,ab]=a[\ell ,b]+[\ell ,a]b$ with the equality established, it is clearly an ideal.

1. $〈[\ell ,S]+S[\ell ,S]〉〈[\ell ,S]+[\ell ,S]S〉=0$. By semi-prime, $[\ell ,S]=0$.

2. $[\ell a,b]=\ell [a,b]+[\ell ,b]a=0$, therefore $\ell S\subseteq \xi \left(S\right)$. And, $[a\ell ,b]=a[\ell ,b]+[a,b]\ell =0$, therefore $S\ell \subseteq \xi \left(S\right)$.

Throughout the rest of this paper, the map ϕ will always refer to a multiplicative CE-epimorphism.

Remark 2.3

Using Lemmas 2.10, 2.11 and 2.12, the ideals $〈{\pi }_{\delta }(x,s)〉$ and $〈{\pi }_{\varphi }(x,s)〉$ for all $x,s\in S$ are central ideals.

Lemma 2.13

Let S be a semi-prime ring. If I is a central ideal in S, then $I\cap 〈[S,S]〉=0$.

Proof.

If $i\in I,$ then $i〈[S,S]〉=〈[iS,S]〉=0$ because $iS\subseteq \xi \left(S\right)$. Since $I\cap 〈[S,S]〉$ is an ideal in S, which squares to zero, $I\cap 〈[S,S]〉=0$.

Lemma 2.14

Let S be a semi-prime ring. Then the set $K=\{k\in S|kS+Sk\subseteq \xi (S\left)\right\}$ satisfies the following.

1. K is an ideal in S.

2. K is a central ideal, i.e. $K\subseteq \xi \left(S\right)$.

Proof.

(1) Let $k\in K$, then $\begin{array}{rl}\left(kS\right)S& =k\left(SS\right)\subseteq \xi \left(S\right).\\ \left(Sk\right)S& =S\left(kS\right)=\left(kS\right)S\subseteq k\left(SS\right)\subseteq \xi \left(S\right).\\ S\left(Sk\right)& \subseteq \left(SS\right)k\subseteq \xi \left(S\right).\end{array}$ Then K is an ideal in S.

(2) For $k\in K$ and $x\in S,$ by the definition of the set K, $[k,x]$ and $[k,x]k$ are elements in $\xi \left(S\right)$. Therefore, $[k,x{]}^2=[[k,x]k,x]-[[k,x],x]k=0$. Using Remark 2.1 and S is a semi-prime ring, we arrive at $[k,x]=0,$ so $K\subseteq \xi \left(S\right)$.

Lemma 2.15

Let S be a two-torsion-free and a 6-centrally torsion-free semi-prime ring. Then the ideal K, defined in Lemma 2.14, satisfies the following.

1. ${\pi }_{\delta }(S,S),{\pi }_{\varphi }(S,S)\subseteq K$.

2. For all $r\in S$ and $k\in K,$ $\delta (r+k)-\delta \left(r\right)\in K$ and $\varphi (r+k)-\varphi \left(r\right)\in K$.

Proof.

(1) Using Lemma 2.10, for all $r,x,t,u\in S$, we have $\left[{\pi }_{\delta }\right(r,x)t,u]={\pi }_{\delta }(r,x)[t,u]=0,$ and $\left[t{\pi }_{\delta }\right(r,x),u]={\pi }_{\delta }(r,x)[t,u]=0.$ Therefore ${\pi }_{\delta }(S,S)\subseteq K$. Similarly, using Lemma 2.11, one can get ${\pi }_{\varphi }(S,S)\subseteq K$.

(2) For $k\in K$, we have $\begin{array}{rl}\delta \left(\right(r+k\left)\right[x,t\left]\right)& =\delta (r+k)[x,t]+(r+k)\delta \left(\right[x,t\left]\right)\\ & +{\pi }_{\delta }(r+k,[x,t\left]\right)\end{array}$ and $\delta \left(r\right[x,t\left]\right)=\delta \left(r\right)[x,t]+r\delta \left(\right[x,t\left]\right)+{\pi }_{\delta }(r,[x,t\left]\right).$ Since $k[x,t]=0,$ by (2) of Lemma 2.14, subtracting gives $\begin{array}{rl}& \left(\delta \right(r+k)-\delta (r\left)\right)[x,t]\\ & =-k\delta \left(\right[x,t\left]\right)-{\pi }_{\delta }(r+k,[x,t\left]\right)+{\pi }_{\delta }(r,[x,t\left]\right)\end{array}$ Thus, using (1) and Lemma 2.13, $\left(\delta \right(r+k)-\delta (r\left)\right)[x,t]\in K\cap 〈[S,S]〉=0$. Thus (14) $\left(\delta \right(r+k)-\delta (r\left)\right)[S,S]=\left\{0\right\}.$(14) Similarly, using analogues steps to the above, we get $\begin{array}{rl}& \delta \left(\right[x,t\left]\right(r+k\left)\right)\\ & =\delta \left(\right[x,t\left]\right)(r+k)+[x,t]\delta (r+k)+{\pi }_{\delta }\left(\right[x,t],r+k),\end{array}$ and $\delta \left(\right[x,t\left]r\right)=\delta \left(\right[x,t\left]\right)r+[x,t]\delta \left(r\right)+{\pi }_{\delta }\left(\right[x,t],r).$ Since $[x,t]k=0,$ by (2) of Lemma 2.14, subtracting gives $\begin{array}{rl}& [x,t]\left(\delta \right(r+k)-\delta (r\left)\right)\\ & =-\delta \left(\right[x,t\left]\right)k-{\pi }_{\delta }\left(\right[x,t],r+k)+{\pi }_{\delta }\left(\right[x,t],r).\end{array}$ Thus, using (1) and Lemma 2.13, $[x,t]\left(\delta \right(r+k)-\delta (r\left)\right)\in K\cap 〈[S,S]〉=0$. Thus (15) $[S,S]\left(\delta \right(r+k)-\delta (r\left)\right)=\left\{0\right\}.$(15) Using (14(14) $\left(\delta \right(r+k)-\delta (r\left)\right)[S,S]=\left\{0\right\}.$(14) ), (15(15) $[S,S]\left(\delta \right(r+k)-\delta (r\left)\right)=\left\{0\right\}.$(15) ), and Lemma 2.12, we arrive at $〈\delta (r+k)-\delta \left(r\right)〉\subseteq \xi \left(S\right)$.

Similar steps to the above proof, using (2) of Lemma 2.14, (1) of Lemmas 2.15, 2.12, and 2.13, one can get the case of epimorphism, that is $〈\varphi (r+k)-\varphi \left(r\right)〉\subseteq \xi \left(S\right)$.

Since K contains all central ideals, $\delta (r+k)-\delta \left(r\right)$ and $\varphi (r+k)-\varphi \left(r\right)$ belong to K.

3. Main results

A mapping $g:S\to S$ preserves the set $A\subseteq S$ if $g\left(A\right)\subseteq A$. It is well known that $\xi \left(S\right)$ is preserved by derivations and epimorphisms. In Ref. [1], the invariance problem for $\xi \left(S\right)$ by CE-derivations and CE-epimorphisms was studied. The authors showed that CE-derivation and CE-epimorphism 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 CE-epimorphism 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 $\xi \left(S\right)$ as well as the ideal K.

Theorem 3.1

Let S be a two-torsion-free and a 6-centrally torsion-free semi-prime ring, and δ, ϕ are multiplicative CE-derivation and multiplicative CE-epimorphism, respectively. Then

1. δ and ϕ preserve the centre $\xi \left(S\right)$, i.e. $\delta \left(\xi \right(S\left)\right)$ and $\varphi \left(\xi \right(S\left)\right)\subseteq \xi \left(S\right)$.

2. δ and ϕ preserve the ideal K, i.e. $\delta \left(K\right)\mathrm{a}\mathrm{n}\mathrm{d}\varphi \left(K\right)\subseteq K$.

Proof.

(1) For $z\in \xi \left(S\right)$ and $\ell \in S$, we have (16) $\delta \left(z\ell \right)=\delta \left(z\right)\ell +z\delta \left(\ell \right)+{\pi }_{\delta }(z,\ell ),$(16) and (17) $\delta \left(\ell z\right)=\delta \left(\ell \right)z+\ell \delta \left(z\right)+{\pi }_{\delta }(\ell ,z).$(17) Subtracting (17(17) $\delta \left(\ell z\right)=\delta \left(\ell \right)z+\ell \delta \left(z\right)+{\pi }_{\delta }(\ell ,z).$(17) ) from (16(16) $\delta \left(z\ell \right)=\delta \left(z\right)\ell +z\delta \left(\ell \right)+{\pi }_{\delta }(z,\ell ),$(16) ), we get $0=\left[\delta \right(z),\ell ]+{\pi }_{\delta }(z,\ell )-{\pi }_{\delta }(\ell ,z),$ which gives $\left[\delta \right(z),\ell ]={\pi }_{\delta }(\ell ,z)-{\pi }_{\delta }(z,\ell )\in \xi \left(S\right).$ However, by Lemma 2.10, $\left[\delta \right(z),\ell {]}^2\in ({\pi }_{\delta }(\ell ,z)-{\pi }_{\delta }(z,\ell )\left)\right[S,S]=(0)$. So, $\left[\delta \right(z),\ell ]$ is a central element that squares to zero. By semi-primeness and Remark 2.1, $\left[\delta \right(z),S]=\left(0\right),$ i.e. $\delta \left(\xi \right(S\left)\right)\subseteq \xi \left(S\right)$.

Similar arguments to the previous case, Lemma 2.11 and Remark 2.1, give us $\varphi \left(\xi \right(S\left)\right)\subseteq \xi \left(S\right)$.

(2) Let $k\in K$ and $a,b\in S$. Then we have (18) $\delta \left(k\right)a=\delta \left(ka\right)-k\delta \left(a\right)-{\pi }_{\delta }(k,a),$(18) and (19) $b\delta \left(k\right)=\delta \left(bk\right)-\delta \left(b\right)k-{\pi }_{\delta }(b,k).$(19) Adding (18(18) $\delta \left(k\right)a=\delta \left(ka\right)-k\delta \left(a\right)-{\pi }_{\delta }(k,a),$(18) ) and (19(19) $b\delta \left(k\right)=\delta \left(bk\right)-\delta \left(b\right)k-{\pi }_{\delta }(b,k).$(19) ) and using part (1) and the definition of K, we get $\delta \left(k\right)a+b\delta \left(k\right)\in \xi \left(S\right)$. Therefore, $\delta \left(K\right)\subseteq K$.

Again, similar arguments to the previous case, give us $\varphi \left(K\right)\subseteq K$.

The following example shows that the conditions two-torsion-free and a 6-centrally torsion-free semi-prime are necessarily in the previous theorem.

Example 3.1

Let $\mathrm{\Re }=(2{\mathbb{Z}}_8,\underset{8}{⨁},\underset{8}{⨀})$ be the sub-ring of the ring of integers $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}8$. $S=\{\left[\begin{array}{ccc}0& r& z\\ 0& 0& u\\ 0& 0& 0\end{array}\right];r,z,u\in \mathrm{\Re }\}$ is a ring with centre $\xi \left(S\right)=\{\left[\begin{array}{ccc}0& r& z\\ 0& 0& u\\ 0& 0& 0\end{array}\right];r,u\in 2\mathrm{\Re }\mathrm{a}\mathrm{n}\mathrm{d}z\in \mathrm{\Re }\}.$ Define the following map $\mathrm{\Omega }:S\to S$ given by $\mathrm{\Omega }\left(\left[\begin{array}{ccc}0& r& z\\ 0& 0& u\\ 0& 0& 0\end{array}\right]\right)=\left[\begin{array}{ccc}0& z& z\\ 0& 0& z\\ 0& 0& 0\end{array}\right].$ Clearly, Ω is a multiplicative CE-derivation (multiplicative CE-endomorphism) and a CE-additive map. A CE-derivation (CE-endomorphism). Also, both $\xi \left(S\right)$ and K are not invariant under Ω.

Using (2) of Lemma 2.15, one can easily show that the maps $\mathrm{\Delta },\mathrm{\Phi }:S/K\to S/K$ that are given by $\mathrm{\Delta }(s+K)=\delta \left(s\right)+K$, and $\mathrm{\Phi }(s+K)=\varphi \left(s\right)+K$ are well-defined. Also, ${\pi }_{\delta }(S,S)\mathrm{a}\mathrm{n}\mathrm{d}{\pi }_{\varphi }(S,S)\subseteq K$, by (1) of Lemma 2.15. So, we arrive at the proof of the following main result.

Theorem 3.2

Let S be a two-torsion-free and a 6-centrally torsion-free semi-prime ring, δ a multiplicative CE-derivation (ϕ a multiplicative CE-epimorphism) on S. Then there exists a central ideal K and a multiplicative derivation $\mathrm{\Delta }:S/K\to S/K$ (a multiplicative epimorphism $\mathrm{\Phi }:S/K\to S/K$) given by $\mathrm{\Delta }(s+K)=\delta \left(s\right)+K$ $\left(\mathrm{\Phi }\right(s+K)=\varphi (s)+K)$.

Example 3.2

Let S be the semi-prime ring $\mathbb{Z}\oplus M_2\left(\mathbb{Z}\right)$. Then $\xi \left(S\right)=\mathbb{Z}\oplus \left[\begin{array}{cc}r& 0\\ 0& r\end{array}\right]$, where $r\in \mathbb{Z}$, and $K=\mathbb{Z}\oplus \left\{0\right\}$. Define the multiplicative CE-derivation $\delta :S\to S$ by $\delta (a\oplus \left[\begin{array}{cc}r& s\\ t& u\end{array}\right])=(a^2+1)\oplus \left[\begin{array}{cc}t-s& u-r\\ r-u& s-t\end{array}\right]$ and the multiplicative CE-epimorphism $\varphi :S\to S$ by $\varphi (a\oplus A)=(a+1)\oplus A;$ $a\in \mathbb{Z}$ and $A\in M_2\left(\mathbb{Z}\right)$. Direct computations give us $S/K\cong M_2\left(\mathbb{Z}\right)$ and $\mathrm{\Delta }:S/K\to S/K$ defined by $\mathrm{\Delta }((0\oplus \left[\begin{array}{cc}r& s\\ t& u\end{array}\right])+K)=(0\oplus \left[\begin{array}{cc}t-s& u-r\\ r-u& s-t\end{array}\right])+K$ is a CE-derivation and $\mathrm{\Phi }:S/K\to S/K$ defined by $\mathrm{\Phi }\left(\right(0\oplus A)+K)=(0\oplus A)+K$ is a CE-epimorphism.

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 (CE-epimorphism ϕ) is a derivation (an epimorphism).

4. Conclusion

Let S be a two-torsion-free and a 6-centrally torsion-free semi-prime ring, and δ, ϕ are multiplicative CE-derivation and multiplicative CE-epimorphism, respectively. Then the set $K=\{r\in S|rS+Sr\subseteq \xi (S\left)\right\}$ is a central ideal and the following results are satisfied

1. δ and ϕ preserve the centre $\xi \left(S\right)$,

2. δ and ϕ preserve the ideal K,

3. There exists a multiplicative derivation $\mathrm{\Delta }:S/K\to S/K$ given by $\mathrm{\Delta }(s+K)=\delta \left(s\right)+K$ and a multiplicative epimorphism $\mathrm{\Phi }:S/K\to S/K$ given by $\mathrm{\Phi }(s+K)=\varphi \left(s\right)+K$.

Acknowledgements

The authors are indebted to Professor I. R. Hentzel, Iowa State University, for his helpful discussions about the idea of this work and his valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).