Exercise 15, § 7

Let A be a ring. A two-sided ideal $\mathfrak{p}$ of A is called prime if for any two-sided ideals $\mathfrak{a}$ and $\mathfrak{b}$ of A, the relation $\mathfrak{a}\mathfrak{b}\subset \mathfrak{p}$ implies $\mathfrak{a}\subset \mathfrak{p}$ or $\mathfrak{b}\subset \mathfrak{p}$.

a) Prove that when A is commutative, this definition coincides with that given in I, §9, No. 3, p. 117.

b) A two-sided ideal $\mathfrak{p}$ of A is prime if and only if for all elements $a$ and $b$ of $A-\mathfrak{p}$, there exists an $x\in A$ such that $axb \notin \mathfrak{p}$.