# Answer to Question #17356 in Abstract Algebra for sanches

Question #17356

Show that, for any direct product of rings Ri, rad ((direct product)Ri) = (direct product) rad Ri.

Expert's answer

Let

*y*= (*yi*)*∈(product)**Ri*. Since*y**∈*rad ((product)*Ri*) amounts to 1*− xy*being left-invertible for any*x*= (*x*)_{i}*∈ (product)**Ri*. This, in turn, amounts to 1*− x*being left-invertible in_{i}y_{i }*Ri*, for any*x*_{i}*∈**Ri*(and any*i*). Therefore,*y**∈*rad ((product)*Ri*) iff*y*_{i}*∈*rad*Ri*for all*i*.
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