Reduced atomic chains of Artinian modules (Q1808750)

If \(R\) is a commutative ring, it is shown that every Artinian module \(A\) has an atomic chain \[ C_A: A= A_0\supset A_1\supset\cdots\supset A_{n-1} \supset A_n=0, \] such that \(\text{Att}_R(A)=\{ \text{Ann}_R(A_{i-1}/A_i: i=1,\ldots,n\}\). Artinian modules \(A\) having an atomic chain of length \(|\text{Att}_R(A)|\) are studied.