Exercise 5, § 1

Let $A$ be a commutative ring in which every ideal is finitely generated. a) An element $p \neq 0$ of $A$ is said to be indivisible if it is not invertible and if the relation $p = xy$ ($x, y \in A$) implies that either $x$ or $y$ belongs to $Ap$. Show that every element $a \neq ()$ of $A$ can be written (in at least one way) as a product of indivisible elements and of one invertible element (use Lemma 1 of VII, p. 4). b) An idempotent $e$ in a commutative ring $A$ is said to be indecomposable if there does not exist a pair of nonzero idempotents $f, g$ such that $f + g = e$ and $fg = 0$. Two distinct indecomposable idempotents have product zero. Show that every idempotent in the ring $A$ is a sum of indecomposable idempotents (show that if e is an idempotent then so is I − e, and use Lemma 1 of VII, p. 4). Deduce that A is the direct product of a finite family of rings $A_i$, in each of which every ideal is finitely generated, and the unit element is the only nonzero idempotent. Suppose this family has at least two elements. Then an element a = $\sum a_i$ (with a, ∈ A, for all i) is indivisible if and only if there is an index k such that a, is either indivisible or zero in $A_k$, and a, is invertible in A, for all i ≠ k ; if $a_k = 0$ then $A_k$ is necessarily an integral domain. c) Show that if 1 is the only nonzero idempotent of A, and if x ∈ A is such that $x^k \in A x^{k-1}$ for some integer k ≥ 1, then either x is invertible or $x^k = 0$ (if $x^k = a x^{k'}$), then consider $a^{k'} x^{k'}$. d) Let p ≠ 0 be a zero divisor in A. Show that there exist a finite sequence $(a_k)$, $1 \leq k \leq m$ of nonzero elements of A such that $0 = pa_1$, $a_k = pa_{k+1}$ for 1 ≤ k < m, $a_m \notin Ap$, and $(a_k) \neq (a_{k+1})$ for 1 ≤ k < m.