Exercise 25, § 3

Let $A$ be a principal ideal ring (Exercise 24), $a$ a two-sided ideal $\neq 0$ of $A$ and $a$ an element of $a$ with the properties stated in Exercise 24 (f). Let $P\{M\}$ denote the following property: $M$ is a left $A$-module and every finitely generated submodule of $M$ is annihilated by a power of $a$.

(a) Show that the conditions (α), (β) and (γ) of Exercise 22 are fulfilled (consider the homothety $a_M$ and argue as in Exercise 23 (a), observing that $\mathrm{Ker}(a_M')$ and $\mathrm{Im}(a_M')$ are submodules of $M$).

(b) Show that every torsion $A$-module $M$ (Exercise 24 (c)) is the direct sum of a submodule $M_a$ which has property $P$ and a submodule $M'_a$ such that the restriction to $M'_a$ of the homothety $a_M$ is bijective (observe that, if $N$ is a torsion $A$-module and $a_N$ is injective, then $a_N$ is bijective; for this, reduce it to the case where $N$ is monogenous and use Exercise 24 (a) and also Algebra, Chapter VIII, § 2, no. 2, Lemma 2).

(c) Let $S$ be the set of elements $s \in A$ whose canonical image in the ring $A/a$ is invertible. Show that $S$ is a multiplicative subset of $A$ and that the following conditions, for any left ideal $I$ of $A$, are equivalent:

(α) $I \cap S \neq \varnothing$; (β) $I + a = A$; (γ) $(A/I)_a = 0$ (in the notation of (b)); (δ) for all $x \in A$, there exists $s \in S$ such that $sx \in I$.

Deduce that $A$ has a (left or right) ring of fractions with respect to $S$ (Chapter 11, § 2, Exercise 22), which is a principal ideal ring and whose only non-zero two-sided ideals are generated by the canonical images of the ideals $a^n$; moreover the canonical mapping of $A$ to this ring of fractions is injective.

(d) Suppose now that the two-sided ideal $a$ is maximal. Show that, for every integer $n > 0$, the ring $A/a^n$ is isomorphic to a matrix ring $\mathbf{M}_r(B_n)$ over a completely primary ring (Algebra, Chapter VIII, § 6, Exercise 20). If $b_n$ is the maximal ideal of $B_n$ show that $b_n^n = 0$, that every (left or right) ideal of $B_n$ is of the form $b_n^k$ and is monogenous. (Note on the one hand that

(Algebra, Chapter VIII, § 6, Exercise 5) and on the other that, for $k < n$, $b_n^k/b_n^{k+1}$ is necessarily a simple $B_n$-module, without which $M_r(b_n^k)$ could not be a monogenous $(A/a^n)$-module.)

Deduce that the completion $\hat{A}$ of $A$ with respect to the topology $T_P(A_s)$ is a matrix ring $M_r(B)$ over a ring $B$ with no divisors of zero and each of whose ideals (left or right) is a power of the same maximal two-sided ideal.