Proof
Part 1.
Part 2.
Since K is a ring, addition on K is a commutative group structure.
Scalar multiplication L × K - > K is given by ordinary multiplication in K. We need to verify the following laws.
-
For x
K:
1x = x;
this clearly holds because of the multiplication laws in K.
-
Associativity: For x, y L
and
z K, we have
x(y(z)) = (xy)z simply because K is associative.
- Distributivity:
For x L
and
y, z K, we have
x(y + z) = xy + xz simply because of distributivity in K.
- Distributivity:
For x, y L
and
z K, we have
(x + y)z = xz + yz again because of distributivity in K.
Linearity of the multiplication with x follows from
- distributivity:
x(y + z) = xy + xz, - and from commutativity:
x( 
y) =
(xy) for
L.