## A Jordan derivation which is not a derivation

Posted: May 8, 2011 in Examples & Counter-Examples, Noncommutative Ring Theory Notes
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Let $R$ be a ring. Recall that an additive map $\delta : R \longrightarrow R$ is called a derivation if

$\delta(r_1r_2)=\delta(r_1)r_2 + r_1 \delta(r_2)$

for all $r_1,r_2 \in R.$ Thus if $\delta$ is a derivation of $R,$ then $\delta(r^2)=\delta(r)r + r \delta(r)$ for all $r \in R.$

Definition. Let $R$ be a ring. An additive map $\delta : R \longrightarrow R$ is called a Jordan derivation if

$\delta(r^2)=\delta(r)r+r\delta(r)$

for all $r \in R.$

So every derivation is a Jordan derivation . But the converse is not true and that is not surprising.

Example. Let $S=\mathbb{C}[x]$ with the relation $x^2=0.$ Let $I=\mathbb{C}x,$ which is an ideal of $S$ because $x^2=0.$ Let

$R = \begin{pmatrix} S & S \\ I & S \end{pmatrix}.$

See that $R,$ with matrix addition and multiplication, is a ring because $I$ is an ideal of $S.$ For any $r = \begin{pmatrix}a & b \\ c & d \end{pmatrix} \in R$ define $\delta(r)=\begin{pmatrix}0 & c \\ 0 & 0 \end{pmatrix}.$ Then $\delta : R \longrightarrow R$ is a Jordan derivation but not a derivation.

Proof. It is obvious that $\delta$ is additive. Let $r = \begin{pmatrix}a & b \\ c & d \end{pmatrix} \in R.$ Then, since $c^2=0,$ we have

$\delta(r^2)=\delta(r)r+r \delta(r) = \begin{pmatrix}0 & (a+d)c \\ 0 & 0 \end{pmatrix}$

and so $\delta$ is a Jordan derivation. To see why $\delta$ is not a derivation, let $r_1=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ and $r_2=\begin{pmatrix}0 & 0 \\ x & 0 \end{pmatrix}.$ Then $\delta(r_1r_2)=0_R$ but $\delta(r_1)r_2 + r_1 \delta(r_2)=\begin{pmatrix}0 & x \\ 0 & 0 \end{pmatrix}. \ \Box$

Is it true that in any ring $R$ with $1,$ if $ab=1$ for some $a,b \in R,$ then $ba=1 ?$
No, that is not true and here’s an example. Let $V$ be a (countably) infinite dimensional vector space over some field $F$ and let $\{v_1,v_2, \cdots \}$ be a basis for $V.$ Now consider $R:=\text{End}_F(V),$ the ring of $F$-linear transformations of $V.$ Define $a,b \in R$ by $a(v_1)=0, \ a(v_j)=v_{j-1}, \ j \ge 2,$ and $b(v_j)=v_{j+1}, \ j \ge 1.$ See that $ab=1$ but $ba \ne 1$ because $ba(v_1)=0.$

Definition 1. A ring $R$ is called Dedekind-finite if $\forall a,b \in R: \ ab=1 \Longrightarrow ba=1.$

Remark 1. Some trivial examples of Dedekind-finite rings: commutative rings, any direct product of Dedekind-finite rings, any subring of a Dedekind-finite ring.

Definition 2. A ring $R$ is called reversible if $\forall a,b \in R : \ ab = 0 \Longrightarrow ba = 0.$

Example 1. Every reversible ring $R$ is Dedekind-finite. In particular, reduced rings are Dedekind-finite.

Proof. Suppose that $ab=1$ for some $a,b \in R.$ Then $(ba-1)b=b(ab)-b=0$ and thus $b(ba-1)=0.$ So $b^2a=b$ and hence $ab^2a=ab=1.$ It follows that $ba=(ab^2a)ba=(ab^2)(ab)a=ab^2a=1.$ So $R$ is Dedekind-finite. Finally, note that every reduced ring is reversible because if $ab=0,$ for some $a,b \in R,$ then $(ba)^2=b(ab)a=0$ and thus $ba=0. \Box$

Example 2. Every (left or right) Noetherian ring $R$ is Dedekind-finite.

Proof. We will assume that $R$ is left Noetherian. Suppose that $ab=1$ for some $a,b \in R.$ Define the map $f: R \longrightarrow R$ by $f(r)=rb.$ Clearly $f$ is an $R$-module homomorphism and $f$ is onto because $f(ra)=(ra)b=r(ab)=r,$ for all $r \in R.$ Now we have an ascending chain of left ideals of $R$

$\ker f \subseteq \ker f^2 \subseteq \cdots.$

Since $R$ is left Noetherian, this chain stabilizes at some point, i.e. there exists some $n$ such that $\ker f^n = \ker f^{n+1}.$ Clearly $f^n$ is onto because $f$ is onto. Thus $f^n(c)=ba-1$ for some $c \in R.$ Then

$f^{n+1}(c)=f(ba-1)=(ba-1)b=b(ab)-b=0.$

Hence $c \in \ker f^{n+1}=\ker f^n$ and therefore $ba-1=f^n (c) = 0. \Box$

Example 3. Finite rings are obviously Noetherian and so Dedekind-finite by Example 2. More generally:

Example 4. If the number of nilpotent elements of a ring is finite, then the ring is Dedekind-finite. See here.

Note that Example 4 implies that every reduced ring is Dedekind-finite; a fact that we proved in Example 1.

Example 5. Let $k$ be a field and let $R$ be a finite dimensional $k$-algebra. Then $R$ is Dedekind-finite.

Proof. Every left ideal of $R$ is clearly a $k$-vector subspace of $R$ and thus, since $\dim_k R < \infty,$ any ascending chain of left ideals of $R$ will stop at some point. So $R$ is left Noetherian and thus, by Example 2, $R$ is Dedekind-finite. $\Box$

Remark 2. Two important cases of Example 5 are $M_n(R),$ the ring of $n \times n$ matrices over a field, and, in general, semisimple rings. As a trivial result, $M_n(R)$ is Dedekind-finite for any commutative domain $R$ because $M_n(R)$ is a subring of $M_n(Q(R))$, where $Q(R)$ is the quotient field of $R$.
So the ring of $n \times n$ matrices, where $n \geq 2,$ over a field is an example of a Dedekind-finite ring which is not reversible, i.e. the converse of Example 1 is not true. Now let $R_i = \mathbb{Z}, \ i \geq 1.$ Then $R= \prod_{i=1}^{\infty} R_i$ is obviously Dedekind-finite but not Noetherian. So the converse of Example 2 is not true.

Example 6 and Example 7 are two generalizations of Example 5.

Example 6. Every algebraic algebra $R$ over a field $k$ is Dedekind-finite.

Proof. Suppose that $ab=1$ for some $a,b \in R.$ Since $R$ is algebraic over $k,$ there exist integers $n \geq m \geq 0$ and some $\alpha_i \in k$ with $\alpha_n \alpha_m \neq 0$ such that $\sum_{i=m}^n \alpha_i b^i = 0.$ We will assume that $n$ is as small as possible. Suppose that $m \geq 1.$ Then, since $ab=1,$ we have

$\sum_{i=m}^n \alpha_i b^{i-1}=a \sum_{i=m}^n \alpha_i b^i = 0,$

which contradicts the minimality of $n.$ So $m = 0.$ Let $c = -\alpha_0^{-1}\sum_{i=1}^n \alpha_i b^{i-1}$ and see that $bc=cb=1.$ But then $a=a(bc)=(ab)c=c$ and therefore $ba=bc=1. \ \Box$

Remark 3. Regarding Examples 5 and 6, note that although any finite dimensional $k$-algebra $R$ is algebraic over $k,$ but $R$ being algebraic over $k$ does not necessarily imply that $R$ is finite dimensional over $k.$ For example, if $\overline{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$ in $\mathbb{C},$ then it is easily seen that $\dim_{\mathbb{Q}} \overline{\mathbb{Q}}=\infty.$ Thus the matrix ring $R = M_n(\overline{\mathbb{Q}})$ is an algebraic $\mathbb{Q}$-algebra which is not finite dimensional over $\mathbb{Q}.$ So, as a $\mathbb{Q}$-algebra, $R$ is Dedekind-finite by  Example 6 not Example 5.

Example 7. Every PI-algebra $R$ is Dedekind-finite.

Proof. Let $J(R)$ be the Jacobson radical of $R.$ If $J(R)=\{0\},$ then $R$ is a subdirect product of primitive algebras $R/P_i,$ where $P_i$ are the primitive ideals of $R.$ Since $R$ is PI, each $R/P_i$ is PI too and thus, by Kaplansky’s theorem, $R/P_i$ is a matrix ring over some division algebra and thus Dedekind-finite by Example 2. Thus $\prod R/P_i$ is Dedekind-finite and so $R,$ which is a subalgebra of $\prod R/P_i,$ is also Dedekind-finite. For the general case, let $S=R/J(R).$ Now, $S$ is PI, because $R$ is PI, and $J(S)=\{0\}.$ Therefore, by what we just proved, $S$ is Dedekind-finite. Suppose that $ab = 1$ for some $a,b \in R$ and let $c,d$ be the image of $a,b$ in $S$ respectively. Clearly $cd=1_S$ and so $dc=1_S.$ Thus $1-ba \in J(R)$ and so $ba=1-(1-ba)$ is invertible. Hence there exists $e \in R$ such that $e(ba)=1.$ But then $eb=(eb)ab=e(ba)b=b$ and hence $ba=(eb)a=e(ba)=1. \Box$

## Central commutators in rings

Posted: August 17, 2010 in Examples & Counter-Examples
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Theorem. Let $R$ be a ring and $a \in R.$ Let $Z(R)$ be the center of $R.$ If $Z(R)$ is reduced and $ar - ra \in Z(R)$ for all $r \in R,$ then $a \in Z(R)$.

Proof. Let $r \in R.$ Then

$ar(ar-ra)=a(ar-ra)r=(a(ar)-(ar)a)r=r(a(ar)-(ar)a)=ra(ar-ra).$

So $(ar-ra)^2=0.$ Hence $ar = ra$ because $Z(R)$ is reduced.  Thus $a \in Z(R). \ \Box$

One class of rings with reduced centers is the class of semiprime rings. If $Z(R)$ is not reduced, the result in the theorem need not hold. There is a nice example in Lam’s book, “A First Course in Noncommutative Rings”. Here it is:

Example. let $k$ be a ring with 1 and let

$R=\left \{ \begin{pmatrix} x & y & z \\ 0 & x & t \\ 0 & 0 & x \end{pmatrix}: \ x,y,z,t \in k \right \}.$

Let $a = e_{12}.$ Then $ar-ra=te_{13} \in Z(R)$ for every $r =\begin{pmatrix} x & y & z \\ 0 & x & t \\ 0 & 0 & x \end{pmatrix} \in R$ but obviously $a \notin Z(R)$ because, for example, $a$ does not commute with $e_{23}.$