$S$-weakly prime ideals in noncommutative rings

UDC 512.5 Let $R$ be a noncommutative ring. The works by Hirano et al. and Groenewald (see [Bull. Korean Math. Soc., 47, No. 5, 1077–1087 (2010)] and [Int. Electron. J. Algebra, 28, 43–68 (2020)]), were devoted to the investigation of weakly prime ideals in noncommutative rings. In this work, we generalize their findings by introducing the concept of $S$-weakly prime ideals via taking $S$ as an $m$-system, which is also a generalization of the concept of right $S$-prime ideals first introduced in [Mediterran. J. Math., 21, No. 2, Article 43 (2024)].