If F is a ﬁeld and R is a ring containing F as a subring, we can just as
easily regard R as a vector space over F as we can in the case that R itself
is ﬁeld. Suppose that R is an integral domain containing the ﬁeld F , and
as a vector space over F , it has ﬁnite dimension. Show that R itself must
be a ﬁeld.
As I understand it, we only need to show that every elements of R have a multiplicative inverse. I don't see how to show that it is true independantly of the definition of multiplication.
Can you help me ?