many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called "noetherian" in her honor. by definition, a noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a noetherian group is defined as a group in which every strictly ascending chain of subgroups is finite. a noetherian module is a module in which every strictly ascending chain of submodules becomes constant after a finite number of steps. a noetherian space is a topological space in which every strictly ascending chain of open subspaces becomes constant after a finite number of steps; this definition makes the spectrum of a noetherian ring a noetherian topological space.