In every local ring exist nonzero nilpotent ideals?

Question

I read that every local ring is clean so there exist clean rings with nonzero nilpotent ideals, i know that a local ring has a unique maximal ideal, but i don't know why do they say this implication.

thank you.

Answer 1

In every local ring exist nonzero nilpotent ideals?

Ridiculous of course, since there are lots of local integral domains.

every local ring is clean so there exist clean rings with nonzero nilpotent ideals, i know that a local ring has a unique maximal ideal, but i don't know why do they say this implication.

Because it is easy to produce local rings with nilpotent ideals, witnessing this statement. $\mathbb Z/ 4\mathbb Z$ is local with a nilpotent maximal ideal. Thus it is local, clean, and has such an ideal.

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Based on the above, it sounds like you misunderstood the strategy of quantification. The claim only asserted “$\exists$ clean ring with nonzero nilpotent ideal” but you interpreted its solution as “$\forall$ local ring, local ring has nonzero nilpotent ideal, hence $\exists$ a clean ring like that.” Really it is already sufficient that “$\exists$ local ring such that it has nonzero nilpotent ideal, hence $\exists$ a clean ring like that.”