Exercise 2, § 8

Let $A$ be a ring. (a) The mapping which associates with $a \in A$ the left homothety $x \mapsto ax$ is an isomorphism of $A$ onto the ring of endomorphisms of the additive group of $A$ which commute with right homotheties. (b) Let $M$ be the set of sequences $(a, b, c, d)$ of four elements of $A$ which may be represented in the form of a table or matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. With every element (a\ b\\c\ d) of M we associate the mapping $(x, y) \mapsto (ax + by, cx + dy)$ of $A \times A$ into itself. Show that a bijection is thus obtained of M onto the ring B of endomorphisms of the commutative group $A \times A$ which commute with the operations $(x, y) \mapsto (xr, yr)$ for all $r \in A$. Henceforth M is identified with the ring B. Calculate the product of two matrices.

(c) Show that the subset T of M consisting of the matrices of the form $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}$ is the subring of M which leaves stable the subgroup $A \times 0$ of $A \times A$.

The mappings $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \mapsto a$ and $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \mapsto c$ are homomorphisms of T into A; let $a$ and $b$ be their kernels. Show that $a + b = T$, that $ab = \{0\}$ and that $ba = b \cap a$ is $\neq \{0\}$ if $A \neq \{0\}$ (cf. no. 9, Proposition 6).

(d) If $A = \mathbf{Z}/2\mathbf{Z}$, T is not commutative, but the multiplicative group of invertible elements of T is commutative.