In this paper, we introduce a new class of ring called [Formula: see text]-[Formula: see text]-Noetherian ring, which is a weak version of [Formula: see text]-Noetherian ring property and study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamation of rings. Furthermore, we define the concept of nonnil [Formula: see text]-[Formula: see text]-Noetherian ring property which is a generalization of the [Formula: see text]-[Formula: see text]-Noetherian domain property and establish a characterization of this notion using pullbacks.

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As in \cite{J}, a ring is called a weak $\pi$-ring if every regular principal ideal is a finite product of prime ideals.In this paper, we establish some characterizations for weak $\pi$-rings. Also, we translate the properties weak $\pi$-ring and $(*)$-ring of $A\propto E$ in terms of a commutative ring $A$ and an $A$-module $E$.

In Section 2:There is a problem in the proof of Lemma 2.4 of the previous version. Indeed, h' is not an S-homomorphism. So, we remove Lemma 2.4 and therefore, assertion 1 of Theorem 2.5, Remark 2.5 and Proposition 2.7 were also removed, as Lemma 2.4 were involved in these results. In Section 3:The assertion 2 of Theorem 3.1 was replaced by Remark 3.6 and therefore Example 3.9 which was using assertion 2 of Theorem 3.1 was removed. In Section 4:In assertion 2 of Theorem 4.1, we have added the condition f(I)J 0 in order to get that the homomorphism π' used in the proof of Lemma 4.3 be well defined.