Hurwitz series over ascending chain of rings

Let [Formula: see text] be an increasing sequence of rings and [Formula: see text]. It is shown that the ring [Formula: see text] (respectively, [Formula: see text]) is Noetherian if and only if (i) the ring [Formula: see text] is Noetherian, (ii) the sequence [Formula: see text] is stationary, (iii) for each [Formula: see text], the [Formula: see text]-module [Formula: see text] is finitely generated and (iv) [Formula: see text]. Also, we show that if [Formula: see text] is a ring extension and [Formula: see text] an ideal of [Formula: see text], the following assertions hold: (1) if the ring [Formula: see text] (respectively, [Formula: see text]) is Noetherian, then (i) the ring [Formula: see text] is Noetherian, (ii) the [Formula: see text]-module [Formula: see text] is finitely generated, (iii) the ideal [Formula: see text] of [Formula: see text] is idempotent and (iv) [Formula: see text]. Conversely, (2) if (i) the ring [Formula: see text] is Noetherian, (ii) the [Formula: see text]-module [Formula: see text] is finitely generated, (iii) the ideal [Formula: see text] of [Formula: see text] is idempotent and (iv) [Formula: see text], then the ring [Formula: see text] (respectively, [Formula: see text]) is Noetherian.