T. Albu and M. L. Teply introduced and investigated [in Contemp. Math. 259, 13-43 (2000; Zbl 0966.06003)] the general concept of a ‘DICC poset’. An arbitrary poset \(P\) is said to be ‘DICC’ or to have the ‘double infinite chain condition’ if any chain of elements of \(P\) indexed by the integers \(\mathbb{Z}\) stabilizes either to the right or to the left, or to both sides; that is, for any chain \[ \cdots\leqslant x_{-2}\leqslant x_{-1}\leqslant x_0\leqslant x_1\leqslant x_2\leqslant\cdots \] of elements of \(P\), there exists \(m\in\mathbb{Z}\) such that \(x_{i+1}=x_i\) for all \(i\geqslant m\) or \(x_{i+1}=x_i\) for all \(i\leqslant m\).
The aim of this paper is to study the DICC for the following two posets of submodules of a right \(R\)-module \(M\): the poset \(\mathcal F\) of all finitely generated submodules of \(M\), and the poset \(\mathcal S\) of all small submodules of \(M\). If one observes that the poset \(\mathcal F\) is Noetherian if and only if \(M\) is a Noetherian module, and the poset \(\mathcal S\) is Artinian if and only if \(M\) is a perfect module, then a series of the author’s results can be immediately deduced from the general properties of DICC posets.