When working with an Artinian ring we must distinguish between being left, right, or two-sided Artinian, but this distinction does not make sense when working with modules.
Unlike the case of rings, there are Artinian modules which are not Noetherian modules. For example, consider the Z-module Q/Z, the additive group of rational numbers modulo the integers. Every submodule of Q/Z is generated by an elements of the form 1/n, for some positive integer n. For any positive integer n greater than one, the chain <1/n>⊂<1/n2>⊂<1/n3>⊂ ... does not terminate, so Q/Z is not Noetherian. Yet every descending chain of submodules terminates: Each such chain has the has the form <1/n1>⊃<1/n2>⊃<1/n3>⊃... for some integers n1, n2, n3, ..., and the inclusion of <1/ni+1> in <1/ni> implies that ni+1 must divide ni, so n1, n2, n3, ... is a decreasing sequence of positive integers. Thus the sequence terminates, making Q/Z is Artinian.
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring.
The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f.
A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R.
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