Clearly it must be (left) noetherian! (Since we wouldn’t want a ascending subnormal chain that doesn’t stop where all of the elements are enjoyed by the larger ideals/submodules) :)
Sorry, writing “R is comm(unitive)” so many times can make you do things
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π is piscist
Surely we step in when someone is descending, right? So our ring is even Artinian
You are an amazing person and I love you, fantastic contribution