On nonnil-$m$-formally Noetherian rings

The purpose of this paper is to introduce a new class of rings containing the class of $m$-formally Noetherian rings and contained in the class of nonnil-SFT rings introduced and investigated by Benhissi and Dabbabi in 2023 \cite{Amir}. Let $A$ be a commutative ring with a unit. The ring $A$ is said to be nonnil-$m$-formally Noetherian, where $m\geq 1$ is an integer, if for each increasing sequence of nonnil ideals $(I_n)_{n\geq 0}$ of $A$ the (increasing) sequence $(\sum_{i_1+\cdots+i_m=n}I_{i_1}I_{i_2}\cdots I_{i_m})_{n\geq 0}$ is stationnary. We investigate the nonnil-$m$-formally Noetherian variant of some well known theorems on Noetherian and $m$-formally Noetherian rings. Also we study the transfer of this property to the trivial extension and the amalgamation algebra along an ideal. Among other results, it is shown that $A$ is a nonnil-$m$-formally Noetherian ring if and only if the $m$-power of each nonnil radical ideal is finitely generated. Also, we prove that a flat overring of a nonnil-$m$-formally Noetherian ring is a nonnil-$m$-formally Noetherian. In addition, several characterizations are given. We establish some other results concerning $m$-formally Noetherian rings.