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Let A be a unital complex Banach algebra and D be a derivation on A. LetK={x ε rad (A): for every natural number n, Dnx ε rad (A)}. Then D preserves the primitive ideals of A invariant if and only if K is closed. We also prove that D maps into the radical if the induced derivation Dk:A/K->A/K is centralising, or K is closed and Dk is spectrally bounded. We also prove some related results.

Let J(R) denote the Jacobson radical of a commutative ring. In Khashan the notion of a J-ideal was introduced and studied. In khashanandece Khashan et al. introduced the concept of weakly J-ideals as a new generalization of J-ideals. In groenewald and groenerwaldweakly it was shown that many of the results are special cases of a more general situation. In TekirEce the notion of an almost n-ideal was introduced and studied. In this note we define almost ρ-ideals for a special radical ρ. If ρ is the prime radical then we have the almost n-radical for NONCOMMUTATIVE rings. We prove amongst other results that an ideal I of a ring is an almost ρ-ideal if and only if I/I² is a weakly ρ-ideal of R. We also investigate rings for which every ideal is an almost radical ideal for a special radical ρ.

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In this paper, we examine the singlet scalar dark matter annihilation to becoming the Standard Model particles in the non-commutative space. In the recent decades, many candidates of dark matter have been offered, but our information about the nature of dark matter is still limited. There are such particle candidates as scalar matetr, fermion, boson, gauge boson, etc.; however, they have neither approved nor rejected. Some complicated models offer many parameters that must be determined. Among these models, the singlet scalar dark matter model is the simplest one that contains just two free parameters to specify. In this paper, we study the singlet scalar dark matter in the NONCOMMUTATIVE space, give the scalar dark matter and higgs lagrangian to find their coupling in NONCOMMUTATIVE space-time and calculate the cross-section of the scalar dark matter pair annihilation to becoming the standard model particles. It must be noted that there is one extra coupling to explain the dark matter annihilation to becoming the standard model particles in the NONCOMMUTATIVE space-time. We investigate the parameter space using the constraint of the thermal average of annihilation cross-section by the velocity. This is the first research on the dark matter annihilation to becoming the standard model particles through the higgs channel in the NONCOMMUTATIVE space.