Exercise 1, § 3

(a) Let $A$ be a ring which is not necessarily commutative and $I$, (resp. $I_d$) the set of elements of $A$ which have no left (resp. right) inverse. Show that the following conditions are equivalent: (1) the sum of two elements of $I$, belongs to $I$; (2) $I$, is a left ideal; (3) there is a greatest left ideal $\mathfrak{J}$ in the set of left ideals $\#A$. If these conditions hold, the opposite ring $A^0$ also satisfies these conditions, $I_1 = I_1 = \mathfrak{J}$, $\mathfrak{J}$ is the unique maximal (left or right) ideal of $A$ and hence the Jacobson radical of $A$, every element $x \notin \mathfrak{J}$ is invertible and $A/\mathfrak{J}$ is a field (cf Algebra, Chapter VIII, § 6, no. 3). Then we also say that $A$ is a local ring.

(b) Show that in a local ring there exists no idempotent other than 0 and 1.