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*_{i}y_{i }being left-invertible in *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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