1 Introduction

Let \(R\) be a ring, not necessarily with 1, with center \(Z=Z(R)\). Let \(N\) be the set of nilpotent elements of \(R\), and call \(R\) reduced if \(N=\{0\}\). For each \(x,y\in R\), denote by \([x,y]\) the commutator \(xy-yx\). If \(S\subseteq R\), define \(f:R\rightarrow R\) to be centralizing on \(S\) if \([x,f(x)]\in Z\) for all \(x\in S\); and define \(f\) to be strong commutativity-preserving on \(S\) if \([x,y]=[f(x),f(y)]\) for all \(x,y \in S\).

Define a map \(D:R\rightarrow R\) to be a centrally-extended derivation (CE-derivation) if for each \(x,y\in R\), \(D(x+y)-D(x)-D(y)\in Z\) and \(D(xy)-D(x)y-xD(y)\in Z\). Define a map \(T:R\rightarrow R\) to be a centrally-extended endomorphism (CE-endomorphism) if for each \(x,y\in R\), \(T(x+y)-T(x)-T(y)\in Z\) and \(T(xy)-T(x)T(y)\in Z\); and if \(T\) is also surjective, call it a CE-epimorphism. We present examples of CE-derivations and CE-endomorphisms and investigate when they are ordinary derivations or endomorphisms, we study their effect on \(Z\), and we note applications to commutativity theorems.

2 Examples and existence theorems

Clearly, every derivation (resp. endomorphism) is a CE-derivation (resp. CE-endomorphism). If \(R\) is commutative, every map \(f:R\rightarrow R\) is both a CE-derivation and a CE-endomorphism; hence we cannot get interesting results in this case.

Example 2.1

Let \(R\) be any ring with \(Z\ne \{0\}\). Choose \(a\in Z\backslash \{0\}\) and define \(T(x)=a\) for all \(x\in R\). Then \(T(x+y)-T(x)-T(y)=-a\in Z\) and \(T(xy)-T(x)T(y)=a-a^2\in Z\), hence \(T\) is a CE-endomorphism. Since \(T\) is not additive, it is not an endomorphism.

Example 2.2

Let \(R\) be a ring with a nonzero central ideal \(I\), and let \(f\) be any function from \(R\) into \(I\).

  1. (a)

    Let \(t\) be any endomorphism of \(R\) and define \(T(x)=t(x)+f(x)\) for all \(x\in R\). Then \(T(x+y)-T(x)-T(y)=f(x+y)-f(x)-f(y)\in Z\) and \(T(xy)-T(x)T(y)=f(xy)-f(x)t(y)-f(y)t(x)-f(x)f(y)\in Z\) for all \(x,y \in R\), so \(T\) is a CE-endomorphism. If \(f\) is a nonzero constant function, \(T\) is not an endomorphism; and if we also take \(t\) to be the identity map, \(T\) is a CE-epimorphism which is not an epimorphism.

  2. (b)

    With \(I\) and \(f\) as above, let \(d\) be a derivation on \(R\) and define \(D(x)=d(x)+f(x)\) for all \(x\in R\). Then \(D\) is easily shown to be a CE-derivation, which for appropriate choices of \(f\) is not a derivation.

Example 2.3

Let \(R_1\) be a commutative domain, \(R_2\) a noncommutative prime ring with derivation \(d\) and \(R=R_1\oplus R_2\). Define \(D:R\rightarrow R\) by \(D((x,y))=(g(x),d(y))\), where \(g:R_1\rightarrow R_1\) is any map which is not a derivation. Then \(R\) is a semiprime ring and \(D\) is a CE-derivation which is not a derivation. Moreover, \(R_1\oplus \{0\}\) is a central ideal of \(R\).

It is no accident that nonzero central ideals play a prominent role in these examples, as the following two theorems show.

Theorem 2.4

Let \(R\) be any ring with no nonzero central ideals. Then every CE-derivation \(D\) on \(R\) is additive.

Proof

Let \(x\) and \(y\) be fixed elements of \(R\), and let

$$\begin{aligned} D(x+y)=D(x)+D(y)+a, \quad a\in Z. \end{aligned}$$
(2.1)

For arbitrary \(t\in R\), we have \(b\in Z\) such that

$$\begin{aligned} D(t(x+y))&=tD(x+y)+D(t)(x+y)+b \nonumber \\&=t(D(x)+D(y)+a)+D(t)(x+y)+b \nonumber \\&=tD(x)+tD(y)+D(t)x+D(t)y+ta+b. \end{aligned}$$
(2.2)

Calculating in a different way, we have

$$\begin{aligned} D(t(x+y))&=D(tx+ty) \nonumber \\&=D(tx)+D(ty)+c \nonumber \\&=tD(x)+D(t)x+b_1+tD(y)+D(t)y+c_1+c, \end{aligned}$$
(2.3)

where \(b_1,c_1,c\in Z\).

Comparing (2.2) and (2.3) gives \(ta+b=b_1+c_1+c\), hence \(Ra\) is a central ideal and therefore \(Ra=\{0\}\). Thus, letting \(A(R)\) be the two-sided annihilator of \(R\), we have \(a\in A(R)\). But \(A(R)\) is a central ideal, so \(a=0\) and by (2.1) \(D(x+y)=D(x)+D(y)\). \(\square \)

Theorem 2.5

If \(R\) is a semiprime ring with no nonzero central ideals, then every CE-derivation \(D\) is a derivation.

Proof

Let \(x,y,t\in R\) be arbitrary elements. Then \(D((xy)t)-xyD(t)-D(xy)t\in Z\) and \(D(x(yt))-xD(yt)-D(x)yt\in Z\). Subtracting, we get

$$\begin{aligned} -xyD(t)-D(xy)t+xD(yt)+D(x)yt\in Z. \end{aligned}$$
(2.4)

Let

$$\begin{aligned} D(xy)&=xD(y)+D(x)y+z_1, \text{ and } \nonumber \\ D(yt)&=yD(t)+D(y)t+z_2, \quad \text{ where } z_1,z_2\in Z. \end{aligned}$$
(2.5)

Then from (2.4),

$$\begin{aligned} -xyD(t)-xD(y)t-D(x)yt-z_1t+xyD(t)+xD(y)t+xz_2+D(x)yt\in Z, \end{aligned}$$

which reduces to

$$\begin{aligned} -z_1t+xz_2\in Z. \end{aligned}$$
(2.6)

Thus \([xz_2,t]=[x,t]z_2=0\). Replacing \(x\) by \(xr, r\in R\), and recalling (2.5), we have

$$\begin{aligned}{}[x,t]R(D(yt)-yD(t)-D(y)t)=0 \quad \text{ for } \text{ all } x,y,t\in R. \end{aligned}$$
(2.7)

Let \(\{P_\alpha {|} \alpha \in \Lambda \}\) be a family of prime ideals of \(R\) such that \(\bigcap P_\alpha =\{0\}\), and let \(P\) denote a typical \(P_\alpha \). Let \(\bar{R}=R/P\) and \(\bar{Z}\) the center of \(\bar{R}\), and let \(\bar{x}=x+P\) be a typical element of \(\bar{R}\).

Fix \(y\) and \(t\) above, and let \(x\) vary. Then \(z_2\) is fixed but \(z_1\) varies with \(x\). Now from (2.7) we have that either \((i) [x,t]\in P\) for all \(x\in R\) or \((ii) z_2=D(yt)-yD(t)-D(y)t\in P\), hence \(\bar{t}\in \bar{Z}\) or \(\bar{z_2}=\bar{0}\). It follows from (2.6) that for each \(x\in R, -\bar{z_1}\bar{t}+\bar{x}\bar{z_2}\in \bar{Z}\), so that if \(\bar{t}\in \bar{Z}, \bar{R}\bar{z_2}\subseteq \bar{Z}\). On the other hand, if \(\bar{z_2}=\bar{0}\), it is certainly true that \(\bar{R}\bar{z_2}\subseteq \bar{Z}\). Thus, \([rz_2,u]\in P\) for all \(r,u\in R\); and since \(\bigcap P_\alpha =\{0\}\), this gives the result that \(Rz_2\) is a central ideal of \(R\). As in the proof of Theorem 2.4, we conclude that \(z_2=0\), i.e., \(D(yt)=yD(t)+D(y)t\). Since \(D\) is additive by Theorem 2.4, our proof is complete. \(\square \)

Combining Theorem 2.5 and Example 2.2(b) gives

Theorem 2.6

A semiprime ring \(R\) admits a CE-derivation which is not a derivation if and only if \(R\) has a nonzero central ideal.

CE-epimorphisms are easily treated by the same methods, so we present our results without proof.

Theorem 2.7

If \(R\) is a ring with no nonzero central ideals, every CE-epimorphism on \(R\) is additive.

Theorem 2.8

If \(R\) is a semiprime ring with no nonzero central ideals, then every CE-epimorphism is an epimorphism.

3 On the invariance problem for \(Z\)

We say that a map \(f:R\rightarrow R\) preserves the subset \(S\subseteq R\) if \(f(S)\subseteq S\). It is well known that derivations and epimorphisms preserve \(Z\), and the purpose of this section is to study preservation of \(Z\) by CE-derivations and CE-epimorphisms.

The CE-derivations of Example 2.2(b) all preserve \(Z\), and so do the CE-epimorphisms of Example 2.2(a) for which \(t\) is an epimorphism. However, there do exist CE-derivations and CE-epimorphisms which do not preserve \(Z\), as the following examples show.

Example 3.1

We give an example of a CE-derivation \(D\) with \(D(Z)\nsubseteq Z\). Let \(R_2\) be a noncommutative ring with \(R^{2}_{2}\subseteq Z(R_2)\), for example a noncommutative ring with \(R_{2}^{3}=\{0\}\). Let \(R_1\) be a zero ring with \((R_1,+)\cong (R_2,+)\) and let \(f:(R_1,+)\rightarrow (R_2,+)\) be an isomorphism. Define \(R\) to be \(R_1\oplus R_2\), and let \(D:R\rightarrow R\) be given by \(D((x,y))=(0,f(x))\). It is easily verified that \(D\) is a CE-derivation on \(R\). Moreover \(R_1\oplus \{0\}\subseteq Z(R)\) and \(D(R_1\oplus \{0\})\nsubseteq Z(R)\).

Example 3.2

Let \(R_1,R_2,f\) and \(R\) be as in Example 3.1. Define \(T((x,y))=(f^{-1}(y),f(x))\) for all \((x,y)\in R\). It is easy to show that \(T\) is a CE-endomorphism; moreover, \(T\) is surjective, since for any \((u,v)\in R, T(f^{-1}(v),f(u))=(u,v)\). Thus, \(T\) is a CE-epimorphism. Again, \(R_1\oplus \{0\}\subseteq Z(R)\) and \(T(R_1\oplus \{0\})\nsubseteq Z(R)\).

Theorem 3.3

Let \(R\) be a ring with \(Z\cap N=\{0\}\). Then every CE-derivation \(D\) on \(R\) preserves \(Z\).

Proof

Let \(z\in Z\) and \(x\in R\). Then \(D(zx)-D(z)x-zD(x)\in Z\) and \(D(xz)-D(x)z-xD(z)\in Z\), and by subtracting we obtain

$$\begin{aligned}{}[x,D(z)]\in Z \quad \text{ for } \text{ all } x\in R. \end{aligned}$$
(3.1)

Replacing \(x\) by \(xD(z)\) in (3.1) gives \([x,D(z)]D(z)\in Z\), so

$$\begin{aligned}{}[[x,D(z)]D(z),x]=0=[x,D(z)]^2 \quad \text{ for } \text{ all } x\in R. \end{aligned}$$
(3.2)

Since \(Z\cap N=\{0\}\), (3.1) and (3.2) give \([x,D(z)]=0\) for all \(x\in R\), i.e., \(D(z)\in Z\). \(\square \)

Theorem 3.4

If \(R\) is a ring with \(Z\cap N=\{0\}\), then every CE-epimorphism on \(R\) preserves \(Z\).

Proof

If \(z\in Z, T(zx)-T(z)T(x)\in Z\) and \(T(xz)-T(x)T(z)\in Z\) for all \(x\in R\), and by subtraction we get \([T(x),T(z)]\in Z\) for all \(x\in R\). Since \(T\) is surjective, this yields \([x,T(z)]\in Z\) for all \(x\in R\). Proceeding as in the previous proof, we get \(T(z)\in Z\). \(\square \)

Corollary 3.5

If \(R\) is a semiprime ring, \(Z\) is preserved by every CE-derivation and by every CE-epimorphism.

CE-derivations and CE-epimorphisms which preserve \(Z\) may also preserve subsets of \(Z\), in particular the set \(K(R)\), defined as \(\{ x\in Z {|} xR\subseteq Z\}\). It is easily shown that \(K(R)\) is a central ideal containing all central ideals, i.e., the maximal central ideal.

Theorem 3.6

If \(D\) is a CE-derivation on a ring \(R\) which preserves \(Z(R)\), then \(D\) preserves \(K(R)\).

Proof

Let \(x\in K(R)\). Since \(K(R)\subseteq Z, D(x)\in Z\). For arbitrary \(r\in R, D(xr)-xD(r)-D(x)r\in Z\); and since \(D(xr)\in Z\) and \(xD(r)\in Z, D(x)r\in Z\). Therefore \(D(x)\in K(R)\). \(\square \)

A similar argument establishes the following theorem.

Theorem 3.7

If \(T\) is a CE-epimorphism on a ring \(R\) which preserves \(Z(R)\), then \(T\) preserves \(K(R)\).

4 Commutativity results

We begin this section with a very easy result.

Theorem 4.1

Let \(R\) be a prime ring and \(D\) (resp. \(T\)) be a CE-derivation (resp. a CE-epimorphism). If \(D(0)\ne 0\) (resp. \(T(0)\ne 0\)), then \(R\) is commutative.

Proof

We give the proof for CE-derivations; the proof for CE-epimorphisms is similar. Let \(D\) be a CE-derivation with \(D(0)\ne 0\). Since \(D(0+0)\) – \(D(0)\) – \(D(0)\in Z\), we have \(D(0)\in Z\). Since \(D(0x)\) – \(D(0)x\) – \(0D(x)\in Z\), we now get \(D(0)x\in Z\) for all \(x\in R\). But \(Z\) contains no nonzero divisors of zero, hence \(x\in Z\) for all \(x\in R\), i.e., \(R\) is commutative. \(\square \)

Theorems 2.6 and 2.8 enable us to replace derivations and epimorphisms by CE-derivations and CE-epimorphisms in certain commutativity theorems. We give two examples.

Theorem 4.2

Let \(R\) be a prime ring and \(U\) a nonzero ideal of \(R\). If \(R\) admits a non-identity CE-epimorphism \(T\) which is strong-commutativity preserving on \(U\), then \(R\) is commutative.

Proof

If \(T\) is an epimorphism, \(R\) is commutative by Bell and Daif (1994), Corollary 2. If \(T\) is not an epimorphism, by Theorem 2.8 \(R\) contains a nonzero central ideal—a condition well known to imply commutativity in a prime ring. \(\square \)

Theorem 4.3

Let \(R\) be a semiprime ring and \(U\) a nonzero left ideal of \(R\). If \(R\) admits a CE-derivation which is nonzero on \(U\) and centralizing on \(U\), then \(R\) contains a nonzero central ideal.

Proof

By Theorem 2.6, \(R\) has a nonzero central ideal or \(D\) is a derivation; and if \(D\) is a derivation, our theorem reduces to Bell and Martindale (1987), Theorem 3. \(\square \)

In general, commutativity theorems with hypotheses involving CE-derivations or CE-epimorphisms seem harder to prove than those with hypotheses involving derivations or epimorphisms. However, there are some possibilities. We conclude with an example, which is a partial generalization of the result that a semiprime ring \(R\) must be commutative if it admits a derivation \(d\) such that \([x,y]=[d(y),d(x)]\) for all \(x,y\in R\). (See Ali and Huang 2012, Theorem 3.3; Liu 2013, Corollary 1.3.)

Theorem 4.4

Let \(R\) be a semiprime ring and \(D\) a CE-derivation on \(R\) such that \([x,y]=[D(y),D(x)]\) for all \(x,y \in R\). If \(R\) is reduced or \(D\) is centralizing on \(R\), then \(R\) is commutative.

Proof

We are assuming

$$\begin{aligned}{}[x,y]=[D(y),D(x)] \quad \text{ for } \text{ all } x,y \in R. \end{aligned}$$
(4.1)

Replacing \(x\) by \(xy\) in (4.1) and using (4.1), we obtain

$$\begin{aligned} D(x)[y,D(y)]+[x,D(y)]D(y)=0 \quad \text{ for } \text{ all } x,y\in R; \end{aligned}$$
(4.2)

and replacing \(x\) by \(yx\) in (4.1), we get

$$\begin{aligned} D(y)[D(y),x]+[D(y),y]D(x)=0 \quad \text{ for } \text{ all } x,y \in R. \end{aligned}$$
(4.3)

Taking \(x=D(y)\) in (4.2) and (4.3), we have for all \(y \in R\)

$$\begin{aligned} D^2(y)[y,D(y)]=[y,D(y)]D^2(y)=0=[D^2(y),[y,D(y)]]. \end{aligned}$$
(4.4)

We now replace \(x\) by \(xw\) in (4.2), thereby obtaining \(z_1\in Z\) such that \((D(x)w+xD(w)+ z_1)[y,D(y)]+[xw,D(y)]D(y)=0\), i.e., \(D(x)w[y,D(y)] + xD(w)[y,D(y)]+z_1[y,D(y)]+x[w,D(y)]D(y)+[x,D(y)]wD(y)=0\); and applying (4.2), we get

$$\begin{aligned} D(x)w[y,D(y)]+z_1[y,D(y)]+[x,D(y)]wD(y)=0. \end{aligned}$$
(4.5)

Taking \(x=D(y)\), we get \(z_2\in Z\) such that

$$\begin{aligned} D^2(y)w[y,D(y)]+z_2[y,D(y)]=0. \end{aligned}$$
(4.6)

It follows that

$$\begin{aligned}{}[D^2(y)w[y,D(y)],[y,D(y)]]=0 \quad \text{ for } \text{ all } y,w\in R, \end{aligned}$$

which reduces for all \(y,w\in R\) to

$$\begin{aligned}{}[D^2(y)w,[y,D(y)]][y,D(y)]&=0, \text{ or } \\ D^2(y)[w,[y,D(y)]][y,D(y)]+[D^2(y),[y,D(y)]]w[y,D(y)]&=0. \end{aligned}$$

Using (4.4), we now get

$$\begin{aligned} D^2(y)w[y,D(y)]^2=0 \quad \text{ for } \text{ all } y,w \in R. \end{aligned}$$
(4.7)

From this equation we obtain

$$\begin{aligned}{}[D^2(y),D(y)]w[y,D(y)]^2=0 \quad \text{ for } \text{ all } y,w \in R, \end{aligned}$$

which by (4.1) is

$$\begin{aligned}{}[y,D(y)]w[y,D(y)]^2=0 \quad \text{ for } \text{ all } y,w \in R; \end{aligned}$$

and invoking semiprimeness of \(R\), we conclude that

$$\begin{aligned}{}[y,D(y)]^2=0 \quad \text{ for } \text{ all } y\in R. \end{aligned}$$
(4.8)

If \(R\) is reduced, it is obvious that \([y,D(y)]=0\); and if \(D\) is centralizing on \(R\), \([y,D(y)]=0\) because \([y,D(y)]\in Z\cap N\). Thus,

$$\begin{aligned}{}[y,D(y)]=0 \quad \text{ for } \text{ all } y\in R. \end{aligned}$$
(4.9)

It follows from (4.5) and (4.9) that \([x,D(y)]w[x,D(y)]=0\) for all \(x,y,w \in R\), hence \(D(R)\subseteq Z\) and therefore \(R\) is commutative by (4.1). \(\square \)