One of these things is called principal if it is generated by a single element.
A closed one of these is called essential if it has nonzero intersection with every other nonzero closed one of these things. If these things are maximal, then they are prime, meaning the set of elements not in them is closed un- der multiplication. Operations on these include   contraction and expansion, which consist of putting them through a homomorphism or through its inverse, respectively. One of these things is called principal if it is generated by a single element. For ten points, identify these subsets of rings which consist of additive groups closed under multiplication.
A closed one of these is called essential if it has nonzero intersection with every other nonzero closed one of these things.
ideals