Exercise 19, § 9

Let $A$ be a non-zero pseudo-ring such that, for all $a \neq 0$ in $A$ and all $b \in A$, the equation $ax + ya = b$ has solutions consisting of ordered pairs $(x, y)$ of elements of $A$; in other words, $aA + Aa = A$ for $a \neq 0$ in $A$.

(a) Show that $A$ is the only non-zero two-sided ideal in $A$.

(b) Show that the relation $a \neq 0$ implies $a^2 \neq 0$ (note that if $a^2 = 0$, it would follow that $aAa = \{0\}$).

(c) Show that if $ab = 0$ and $a \neq 0$ (resp. $b \neq 0$) then $b = 0$ (resp. $a = 0$). (Note that $(ba)^2 = 0$ and deduce that $bAb = \{0\}$.)