文档介绍：page 1 of Chapter 4
CHAPTER 4 MODULE FUNDAMENTALS
Modules and Algebras
Definitions ments
A vector space M over a field R is a set of objects called vectors, which can be added,
subtracted and multiplied by scalars (members of the underlying field). Thus M is an
abelian group under addition, and for each r ∈ R and x ∈ M we have an element rx ∈ M.
Scalar multiplication is distributive and associative, and the multiplicative identity of the
field acts as an identity on vectors. Formally,
r(x + y)=rx + ry;(r + s)x = rx + sx; r(sx)=(rs)x;1x = x
for all x, y ∈ M and r, s ∈ R. A module is just a vector space over a ring. The formal
definition is exactly as above, but we relax the requirement that R be a field, and instead
allow an arbitrary ring. We have written the product rx with the scalar r on the left, and
technically we get a left R-module over the ring R. The axioms of a right R-module are
(x + y)r = xr + yr; x(r + s)=xr + xs;(xs)r = x(sr),x1=x.
“Module” will always mean left module unless stated otherwise. Most of the time, there is
no reason to switch the scalars from one side to the other (especially if the underlying ring
mutative). But there are cases where we must be very careful to distinguish between
left and right modules (see Example 6 of ()).
Some Basic Properties of Modules
Let M be an R-module. The technique given for rings in () can be applied to
establish the following results, which hold for any x ∈ M and r ∈ R. We distinguish the
zero vector 0M from the zero scalar 0R.
(1) r0M =0M [r0M = r(0M +0M )=r0M + r0M ]
(2) 0Rx =0M [0Rx =(0R +0R)x =0Rx +0Rx]
(3) (−r)x = r(−x)=−(rx) [as in (2) of () with a replaced by r and b by x]
(4) If R is a field, or more generally a division ring, then rx =0M implies that either r =0R
−1
or x =0M . [If r = 0, multiply the equation rx =0M by r .]
Examples
1. If M is a vector space over the field R, then M is an R-module.
2. Any ring R is a module over itsel