2.3 Non-commutative rings

In Algebra I and II, no serious non-commutative ring theory is covered, which is the main subject of Algebra III. A few very basic notions you should know anyhow.

Know basic ring definitions: Ring, subring; (group of) units, divison ring (or skew field); zero divisors; left-, right- and two-sided ideals; quotient module $R / I$ for a left ideal $I$ and quotient ring $R / I$ for a two-sided ideal $I$; (side remark: the existence of maximal left-, right- and two-sided ideals again follows from Zorn’s Lemma); ring homomorphisms and isomorphisms; center $Z(R)$ of a ring $R ; K$-algebra for a field $K$.

Know two basic examples: the Hamiltonian quaternions $\mathbb{H}=\mathbb{R} \oplus \mathbb{R} i \oplus \mathbb{R} j \oplus \mathbb{R} k$; the matrix algebra $M_{n}(K)$ for a field $K$.

Related Problems

( $R$ here is a not necessarily commutative ring with unit 1 )

1. (Nov $77 \# 10$ ) Let $F$ be a field of characteristic $p>0$. A $p$-polynomial is a polynomial $f(x)=a_{n} x^{p^{n}}+\ldots+a_{1} x^{p^{1}}+a_{0} x^{p^{0}}=a_{n} x^{p^{n}}+\ldots+a_{1} x^{p}+a_{0} x$ which is a linear combination of $p^{e}$-th powers of $x(e=n, . ., 1,0)$. If $a_{n} \neq 0$ then $f$ has degree $n$.
   (a) Show that the set $R$ of all $p$-polynomials becomes a non-commutative ring under the usual addition and the substitution product $f(x) * g(x)=f(g(x))$. Does $R$ have a unit element? What are the zero divisors?
   (b) Show that every left ideal $I$ in $R$ is principal, $I=R f$.
   (c) Show that every right ideal of $R$ is principal iff the field $F$ is perfect ( $F^{p}=F$ ).
   (d) Are there any perfect fields $F$ for which $R$ is commutative?

2. (Sep $84 \# 2$ ) Let $L$ be a left ideal in $R$.
   (a) Show $I(L)=\{a \in R \mid L a \subseteq L\}$ is the largest subring $S$ of $R$ such that $L$ is a 2-sided ideal of $S$.
   (b) Prove that $R$ is a division ring iff $I(L)=L$ for all nonzero $L$.

3. (1985#3d) An element of $R$ is nilpotent if $x^{n}=0$ for some $n$. Show that if $x, y$ are commuting nilpotent elements in $R$ then so is $x+y$; give an example to show this is not true if $x, y$ do not commute.

4. (Sep 86 #7) (a) Define the quaternions $\mathbb{H}$ over the reals.
   (b) Show that any homomorphism of $\mathbb{H}$ into the complex numbers is identically zero.
   (c) Prove that the equation $x^{2}+1=0$ has infinitely many solutions in $\mathbb{H}$. Why can’t you deduce from the "factorization" $x^{2}+1=(x+i)(x-i)$ and the fact that $\mathbb{H}$ is a division ring that there are only two solutions?
   (d) How many solutions has the equation $x^{2}-1=0$ in $\mathbb{H}$ ?

5. (May 89 #2) Show that in a general ring (not necessarily commutative or with unit), all the elements that are not divisors of zero have the same additive order. What are the possible values for this order?

6. (Aug $89 \# 2$ ) If $(a+b)^{2}=a^{2}+b^{2}$ for all $a, b$ in $R$, show $R$ is commutative.

7. (Jan 94 #5) Give an example (and a proof that it works) of a ring $R$ without identity and an ideal in the ring direct sum $R \oplus R$ that does not have the form $I_{1} \oplus I_{2}$ where the $I_{k}$ are ideals in $R$.

8. (Aug 98 #5) A derivation $D$ of a ring $R$ is a map of $R$ into itself such that $D(a+b)= D(a)+D(b)$ and $D(a b)=D(a) b+a D(b)$ for all elements $a, b$ of $R$. Show that if $D$ is a derivation, and in addition $D^{2}=0$ and $R$ has no 2-torsion ( $2 a=0$ implies $a=0$ ), then the "exponential map" I $d+D$ is an automorphism of $R$.

9. (Jan 05 #4) Let $K$ be a field and $A$ a finite-dimensional $K$-algebra (that is $A$ is a ring with 1 containing $K$ in its center with $\operatorname{dim}_{K} A<\infty$ ). Show that any element of $A$ is either a zero divisor or a unit.

10. (Aug $10 \# 7$ ) Let $K$ be a field. Let $A$ be a finite-dimensional (possibly non-commutative) $K$-algebra (with 1 ), and assume that $A$ is a division ring.
    (a) Prove that every $K$-subalgebra of $A$ is a division ring.
    (b) Assume that $K$ is algebraically closed. Prove that $\operatorname{dim}_{K} A=1$.

11. (Aug $11 \# 1$ ) Let $M$ be a simple (left) module for a ring $R$. This means that $M$ has no submodules apart from 0 and $M$.
    (a) Prove that $M \cong R / I$ where $I$ is a maximal left ideal of $R$.
    (b) Show that $E:=\operatorname{End}_{R}(M)$ is a division ring, i.e., every nonzero element of $E$ is invertible.

12. (Jan $12 \# 4$ ) Let $R$ be a ring with 1 . The opposite ring $R^{o p}$ is defined as follows: as a set $R^{o p}=R$, the addition on $R^{o p}$ coincides with the addition on $R$ and the multiplication $*$ on $R^{o p}$ is the multiplication on $R$ in reverse order, that is, $a * b=b a$ (where $b a$ is the product in $R)$. Let $e \in R$ be an idempotent element, that is, $e^{2}=e$.
    (a) Prove that $e R e=\{$ ere $: r \in R\}$ is a ring with multiplicative identity $e$.
    (b) Consider the left R-module M = Re. Prove that its endomorphism ring $\operatorname{End}_{R}(M)= \operatorname{Hom}_{R}(M, M)$ is isomorphic to $(e R e)^{o p}$, the opposite ring of $e R e$.

13. (Aug 14 #8) Let $A=M_{n}(R)$ be the algebra of $n \times n$ matrices over a commutative ring $R$ with 1 . Fix $1 \leq i, j \leq n$. Determine
    (a) the left ideal of $A$ generated by $E_{i j}$;
    (b) the (two-sided) ideal of $A$ generated by $E_{i j}$;
    (c) Are there possibly other nonzero ideals of $A$ besides those of the form (b)?
    (Here as usual $E_{i j}$ denotes the matrix whose $(i, j)$ th entry is 1 and 0 elsewhere.)

14. (Jan 17 #8) Let $A$ be the ring of $n \times n$ matrices over a field $F$.
    (a) Show the right ideals of $A$ are precisely the subsets of the form

$$\{X \in A \mid \operatorname{image}(X) \subset V\}$$

where $V$ ranges over all linear subspaces of $F^{n}$.
(b) Show the left ideals of $A$ are precisely the subsets of the form

$$\{X \in A \mid \operatorname{kernel}(X) \supset W\}$$

where $W$ ranges over all linear subspaces of $F^{n}$.
(c) Show that $A$ is a simple ring: its only 2 -sided ideals are $A$ itself, and $\{0\}$.

Chapter 3

Modules and canonical forms