For ten points, identify these algebraic structures with an additive and multiplicative operation, examples of which include the one of integers.
Modules are like vector spaces except that their coe cients are taken from these and not  elds. These entities are called Artinian if they ful ll the de- scending   chain condition, and the octonions form a non-associative one of these entities. A commutative one of these entities with a unit element and no divisors of zero is called an integral domain. These entities are referred to as simple if their only two-sided ideals are themselves and zero. Unlike groups, these objects have two binary operations, only one of which requires an identity or inverses. For ten points, identify these algebraic structures with an additive and multiplicative operation, examples of which include the one of integers.
Modules are like vector spaces except that their coe cients are taken from these and not  elds.
rings