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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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Let R be a commutative ring with nonzero identity. Yassine et al. defined the concept of 1-absorbing prime ideals as follows: a proper ideal I of R is said to be a 1-absorbing prime ideal if whenever xyz 2 I for some nonunit elements x, y, z 2 R, then either xy 2 I or z 2 I: We use the concept of 1absorbing prime ideals to study those commutative rings in which every proper ideal is a product of 1-absorbing prime ideals (we call them OAFrings). Any OAF-ring has dimension at most one and local OAF-domains (D, M) are atomic such that M 2 is universal.