In the highly respectable BBFSK Vol I B2 we are told that the integers form a module with respect to addition. Where by module they mean a set together with an operation (+) defined in it satisfying the requirement of
- associativity and commutativity,
- a neutral element applicable to every element,
- an inverse to every element.
In a footnote they say this is also called a commutative (or Abelian) group.
That is not how I would typically enumerate the laws/axioms/properties of an Abelian group, but it is consistent with the familiar definition. The term module is otherwise unfamiliar to me. I've been told that the use of the term in this way is non-standard. Is that the case? If so, how is this similar and different from the currently accepted definition?
Best Answer
"Nowadays", and for as long as I have known, a module over a ring is a notion that generalises that of a vector space over a field.
Since apparently a vector space is an abelian group, well, there you have it.