Exercise 12, § 5

Let A be a ring, I an A-module, B the unique ring whose underlying additive group is A ⊕ I, A being a subring of B, I a square zero ideal, whose A-module structure obtained by restriction of scalars is the given structure (Alg., chap. II, 3rd ed., § 1, exerc. 7).

a) Suppose that every noninvertible element of A belongs to the annihilator of an element ≠ 0 of I. Show that in B every element which is not a divisor of 0 is invertible, in other words that B is equal to its total ring of fractions.

b) Let $(m_\lambda)_{\lambda \in L}$ be a family of maximal ideals of A such that every noninvertible element of A belongs to at least one $m_\lambda$. Show that, if one takes for I the direct sum of the A-modules $A/m_\lambda$, the condition of a) is satisfied.

c) Deduce from b) that, if there exists a projective A-module of rank 1 which is not free, one can determine I so that B is equal to its total ring of fractions, but that there exists a projective B-module of rank 1 which is not free (this module being necessarily noninvertible, since B is the only nondegenerate sub-B-module of B).

d) Suppose that there exists in A a maximal ideal m such that, for every $x \in m$, there exists a maximal ideal $m' \neq m$ of A containing x. If one takes for I the direct sum A-module of the $A/m'$, where $m'$ runs through the set of maximal ideals of A distinct from m, the corresponding ring $B = A \oplus I$ is equal to its total ring of fractions; if one puts $n = m \oplus I$, which is a maximal ideal of B, show that $B_n$ is isomorphic to $A_m$; if A is integral (necessarily different from a field by virtue of the assumption), $B_n$ is therefore an integral ring distinct from its field of fractions. If moreover mAₘ is a principal ideal of the integral ring $A_m$, show that the ideal $mB = m \otimes_A B$ of B is a degenerate nonfree projective B-module of rank 1. (One will see in chap. VII that there are Dedekind rings A in which there exist maximal ideals m of which no power is a principal ideal; for such a ring A, all the preceding assumptions are satisfied.)