Among modules over a commutative ring , the module has a nice feature: homomorphisms from to another module are entirely determined by where they send the elements , , up to . Moreover, any choice of where we want those basis elements to be sent in extends uniquely to a homomorphism . Thus homomorphisms correspond bijectively to -tuples of elements of . This is an example of a universal property: is the universal module equipped with an -tuple of elements, also called the free module on elements.
What about the free module on an infinite collection of elements? Continue reading