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
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