Rings and Their Homomorphisms

Let $M_{2 \times 2}$ be the ring of real $2 \times 2$ matrices and $S \subset M_{2 \times 2}$ the subring of matrices of the form

Let $M_{2 \times 2}(\mathbb {Q})$ denote the ring of $2 \times 2$ matrices with entries in $\mathbb {Q}$. Let $R$ be the set of matrices in $M_{2 \times 2}(\mathbb {Q})$ that commute with $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.

Prove that there exists only one automorphism of the field of real numbers; namely the identity automorphism.

Suppose addition and multiplication are defined on $\mathbb {C}^n$, complex $n$-space, coordinatewise, making $\mathbb {C}^n$ into a ring. Find all ring homomorphisms of $\mathbb {C}^n$ onto $\mathbb {C}$.

Let $R$ be a finite ring. Prove that there are positive integers $m$ and $n$ with $m > n$ such that $x^m = x^n$ for every $x$ in $R$.

Let $R$ be a ring with at least two elements. Suppose that for each nonzero $a$ in $R$ there is a unique $b$ in $R$ (depending on $a$) with $aba = a$. Show that $R$ is a division ring.

Let $p$ be a prime number and $R$ a ring with identity containing $p^2$ elements. Prove that $R$ is commutative.

Let $R$ be a ring with identity, having fewer than eight elements. Prove that $R$ is commutative.

Let $R$ be the set of complex numbers of the form

Prove that $R$ is a subring of $\mathbb {C}$, and that $R$ is an integral domain but not a unique factorization domain.

Let $R$ be a commutative ring with identity. Let $G$ be a finite subgroup of $R^{*}$, the group of units of $R$. Prove that if $R$ is an integral domain, then $G$ is cyclic.

Let $R$ be a finite ring with identity. Let $a$ be an element of $R$ which is not a zero divisor. Show that $a$ is invertible.

Let $R$ be a ring with identity, and let $u$ be an element of $R$ with a right inverse. Prove that the following conditions on $u$ are equivalent:

1. $u$ has more than one right inverse;

2. $u$ is a zero divisor;

3. $u$ is not a unit.

Show that no commutative ring with identity has additive group isomorphic to $\mathbb {Q} / \mathbb {Z}$.

Let $D$ be an ordered integral domain and $a \in D$. Prove that

Prove that $\mathbb {Q}^{x,y} / \left\langle x^2+y^2-1\right\rangle$ is an integral domain and that its field of fractions is isomorphic to the field of rational functions $\mathbb {Q}(t)$.

Find the cardinality of the set of all subrings of $\mathbb {Q}$, the field of rational numbers.