# Answer on Algebra Question for Mohammad

Question #17361

Show that a ring R is von Neumann regular iff IJ = I ∩ J for every right ideal I and every left ideal J in R.

Expert's answer

First assume

*R*is von Neumannregular. For*I, J*as above, it suffices to show that*I ∩ J**⊆**IJ*. Let*a**∈**I ∩ J*. There exists*x**∈**R*such that*a*=*axa*. Thus,*a**∈**(**IR*)*J**⊆**IJ*.Conversely, assume that*IJ*=*I ∩ J*for any right ideal*I*andany left ideal*J*. For any*a**∈**R*, we have then*a**∈**(**aR*)*∩*(*Ra*) = (*aR*)(*Ra*) =*aRa.*Need a fast expert's response?

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