Exercise 7, § 1

(a) Let A be a ring and E an (A, A)-bimodule; on the product $B = A \times E$ a ring structure is defined by setting

E (canonically identified with $\{0\} \times E$) is then a two-sided ideal of B such that $E^2 = \{0\}$.

(b) By suitably choosing A and E, show that E can be a left B-module with $\gamma(E)$ (Exercise 6) an arbitrary cardinal, although $\gamma(B_s) = 1$.