Exercise 15, § 8

A ringoid is a set $E$ with two laws of composition: (a) an associative multiplication $xy$; (b) a law written additively, not everywhere defined (§ 1, no. 1) and satisfying the following conditions:

(1) it is commutative (in other words, if $x + y$ is defined, so is $y + x$ and $x + y = y + x$; $x$ and $y$ are then said to be addible);

(2) if $x$ and $y$ are addible, for $x + y$ and $z$ to be addible it suffices that $x$ and $y$ on the one hand, and $y$ and $z$ on the other, be addible; then $x$ and $y + z$ are addible and $(x + y) + z = x + (y + z)$;

(3) there exists an identity element 0;

(4) if $x$ and $z$ on the one hand, $y$ and $z$ on the other, are addible and $x + z = y + z$, then $x = y$;

(5) if $x, y$ are addible, so are $xz$ and $yz$ (resp. $zx$ and $zy$) and

(resp. $z(x + y) = zx + zy$) for all $z \in E$.

Every ring is a ringoid.

(a) Examine how the definitions and results of § 8 and the above exercises extend to ringoids (a left ideal of a ringoid $E$ is a subset $a$ of $E$ which is stable under addition and such that $E.a \subset a$).

(b) Let $G$ be a group with operators and $f$ and $g$ two endomorphisms of $G$. For the mapping $x \mapsto f(x)g(x)$ to be an endomorphism of $G$, it is necessary and sufficient that every element of the subgroup $f(G)$ be permutable with every element of $g(G)$; if this endomorphism is then denoted by $f + g$ and $fg$ is the composite endomorphism $x \mapsto f(g(x))$, show that the set $E$ of endomorphisms of $G$ with these laws of composition is a ringoid with unit element; for $E$ to be a ring, it is necessary and sufficient that $G$ be commutative.

For an element $f \in E$ to be addible to all the elements of $E$, it is necessary and sufficient that $f(G)$ be contained in the centre of $G$; the set $N$ of these endomorphisms is a pseudo-ring called the kernel of the ringoid $E$.

An endomorphism $f$ of $G$ is called normal if it is permutable with all the inner automorphisms of $G$; for every normal stable subgroup $H$ of $G$, $f(G)$ is then a normal stable subgroup of $G$. Show that the set $D$ of normal endomorphisms of $G$ forms a subringoid of $E$ and that the kernel $N$ is a two-sided ideal in $D$.†