One measure of the dimensionality of these objects is the least upper bound on the heights of its closed, additive group-forming sub-modules. The unmixedness theorem holds for examples of these objects in which the aforementioned dimensionality measure is equal to the depth. That dimensionality measure is the Krull dimension, which is equal to the depth in the examples of these named for Cohen and Macaulay. The major classifications of these objects are in terms of chain conditions. The aforementioned closed, additive group-forming sub-modules of these objects are know as prime (*) ideals; an ideal is a sub-module of one of these that commutes with all its elements. These algebraic structures are closed under both an associative, commutative addition, identity-bearing invertable addition, and a left- and right-distributive, associative multiplication. For 10 points, name these algebraic structures that may be Artanian or Noetherian.
One measure of the dimensionality of these objects is the least upper bound on the heights of its closed, additive group-forming sub-modules. The unmixedness theorem holds for examples of these objects in which the aforementioned dimensionality measure is equal to the depth. That dimensionality measure is the Krull dimension, which is equal to the depth in the examples of these named for Cohen and Macaulay. The major classifications of these objects are in terms of chain conditions. The aforementioned closed, additive group-forming sub-modules of these objects are know as prime (*) ideals; an ideal is a sub-module of one of these that commutes with all its elements. These algebraic structures are closed under both an associative, commutative addition, identity-bearing invertable addition, and a left- and right-distributive, associative multiplication. For 10 points, name these algebraic structures that may be Artanian or Noetherian.
One measure of the dimensionality of these objects is the least upper bound on the heights of its closed, additive group-forming sub-modules.
{ring}s [prompt on {field}s or {module}s]