noetherian rings redux


i've been going over some problems on noetherian rings
because i have noticed i am really lacking in my abilities to work with them
and given their importance almost everywhere these days i needed to sit down and practice

i just worked through this problem:

Given a commutative noetherian ring R, show that

oo
- i
| | radical(R) = 0
i=1

where the symbol indicates intersection.

this problem turned out to be a consequence of krull's intersection theorem for integral domains
but the problem also said this is not true in general if:
R is noncommutative and only left noetherian (or only right noetherian)

that seemed a weird wording to me

is this true if R is both left and right noetherian or not?

if it is true
what does a sketch of the proof look like (or lit refs are fine)?

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galathaea: prankster, fablist, magician, liar
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