On right $S$-$(1,\mathcal{P})$-absorbing ideals in noncommutative rings
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Abstract
We introduce and investigate the class of right $S$-$(1,\mathcal{P})$-absorbing ideals in noncommutative rings, which extends the notion of strongly $S$-$1$-absorbing primary ideals from the commutative setting. An ideal $K$ of a ring $R$ disjoint from an $m$-system $S$ is called right $S$-$(1,\mathcal{P})$-absorbing if, whenever $a,b,c \in R$ are nonunits with $aRbRc \subseteq K$, then either $ab\langle s\rangle \subseteq K$ or $c\langle s\rangle \subseteq \mathcal{P}(R)$ for some $s \in S$. We establish fundamental properties of these ideals, explore their connections with right $S$-prime and $S$-$\mathcal{P}$-ideals, and present illustrative examples. Structural results concerning localization, homomorphisms, and trivial ring extensions are obtained. In particular, we show conditions under which right $S$-$(1,\mathcal{P})$-absorbing ideals coincide with $S$-primary ideals, and we characterize their behavior in local rings. These results demonstrate how right $S$-$(1,\mathcal{P})$-absorbing ideals provide a natural framework unifying several prime-like generalizations.
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