On weakly semiprime ideals in noncommutative ring

Abstract

We extend the concept of weakly semiprime ideals, originally defined by A. Badawi for commutative rings, to the noncommutative setting. We define a proper ideal I of a noncommutative ring R to be weakly semiprime if for any a ∈ R, 0 ≠ aRa ⊆ I implies a ∈ I. Fundamental properties of such ideals are investigated, particularly those that are weakly semiprime but not semiprime. Key results include showing that such ideals are contained in the prime radical of the ring. These ideals are also characterized in decomposable rings (direct products), and a method for constructing examples using the idealization of a bimodule is provided. We further introduce and study related concepts, including "ideal weakly semiprime" ideals (where 0 ≠ A² ⊆ I for an ideal A implies A ⊆ I), weakly completely semiprime ideals, and an extension of these notions to submodules of an R-module.