Hilbert's Basis Theorem provides a condition for when one of these is Noetherian, which implies that the ascending chain condition is satisfied by its special subsets, known as ideals. A generalization to a vector space known as a module makes use of this structure, and homomorphisms between these objects preserve both of the characteristic operations. Though they are not generally commutative, the most interesting varieties such as integral domains and unique factorization domains are. For ten points, identify this algebraic structure more general than a field that possesses addition and multiplication, but does not 
Hilbert's Basis Theorem provides a condition for when one of these is Noetherian, which implies that the ascending chain condition is satisfied by its special subsets, known as ideals. A generalization to a vector space known as a module makes use of this structure, and homomorphisms between these objects preserve both of the characteristic operations. Though they are not generally commutative, the most interesting varieties such as integral domains and unique factorization domains are. For ten points, identify this algebraic structure more general than a field that possesses addition and multiplication, but does not necessarily have multiplicative inverses.
Hilbert's Basis Theorem provides a condition for when one of these is Noetherian, which implies that the ascending chain condition is satisfied by its special subsets, known as ideals.
{ring}