cardinalities of bases for modules
Let be a ring and a left module over .
If has a finite basis, then all bases for are finite.
Suppose is a finite basis for , and is another basis for . Each element in can be expressed as a finite linear combination of elements in . Since is finite, only a finite number of elements in are needed to express elements of . Let be this finite subset (of ). is linearly independent because is. If , pick . Then is expressible as a linear combination of elements of , and subsequently a linear combination of elements of . This means that , or , contradicting the linear independence of . ∎
Suppose be a basis for with , the smallest infinite cardinal, and is another basis for . We want to show that . First, notice that by the previous proposition. Each element can be expressed as a finite linear combination of elements of , so let be the collection of these elements. Now, is uniquely determined by , as is a basis. Also, is finite. Let
Since spans , so does . If , pick , so that is a linear combination of elements of . Moving to the other side of the expression and we have expressed as a non-trivial linear combination of elements of , contradicting the linear independence of . Therefore . This means
Similarly, every element in is expressible as a finite linear combination of elements in , and using the same argument as above,
|Title||cardinalities of bases for modules|
|Date of creation||2013-03-22 18:06:33|
|Last modified on||2013-03-22 18:06:33|
|Last modified by||CWoo (3771)|