Definition talk:Standard Ordered Basis

Re the Definition:Standard Ordered Basis I wrote, I don't know if the same ambiguity is a problem with rings in general, because I haven't studied rings. Is there a similar issue with ring elements? --GFauxPas 10:13, 6 May 2012 (EDT)
 * The issue is to do with the order in which the elements are stated.
 * Such a concept appears only to be defined only on a Module on Cartesian Product, whose factors are specific instances of a given ring with unity. As a Cartesian product consists by definition of ordered tuples, it is possible to create an ordered basis as defined here: consisting of the unity of the $i$th element and the ring zero of the non-$i$th elements.
 * If the module on which you are creating a basis is not specifically ordered (like your one in the example), you can't create a standard ordered basis for it. By definition, the example given is not a standard ordered basis. --prime mover 10:56, 6 May 2012 (EDT)
 * Ah, okay. Maybe it's worth a page that proves that the set of matrices of a certain size with one non-zero entry and the rest zero entries are bases of the corresponding matrix space, though. --GFauxPas 11:10, 6 May 2012 (EDT)
 * Probably, but I think it's worth writing the page that defines "standard basis" first, so as to make it clear that it's a different entity from a "standard ordered basis". I'd be advised to wait till you've got more of the theory under your belt (ring theory and field theory for example, i.e. wait till you're more familiar with what "abstract algebra" is really all about). Just mu opinion, you may believe otherwise. --prime mover 11:21, 6 May 2012 (EDT)