Exercise 4, § 7

Let A be a primitive ring.

a) If the ring A is commutative (resp. left Artinian), then it is a field (resp. a simple ring).

b) Let $\mathfrak{a}$ and $\mathfrak{b}$ be nonzero left ideals of A. Prove that the ideal $\mathfrak{a}\mathfrak{b}$ is not zero.[^1] Deduce that the intersection of finitely many nonzero two-sided ideals of A is not reduced to 0.

c) Prove that if for every $a\in A$, the element $1 +a^2$ is invertible in A, then A is a field (use Exercise 3, b)).

[^1]: In this exercise and the next, if $\mathfrak{a}$ and $\mathfrak{b}$ are left ideals of A, we denote by $\mathfrak{a}\mathfrak{b}$ the left ideal generated by the products $ab$ for $a\in \mathfrak{a}$ and $b\in \mathfrak{b}$.