The letters F, K, L will always denote fields.
Definition. Let K L be fields, L. Let K[x] L be the canonical map induced by
sending x 7 . If this homomorphism is injective, it extends to a mapping K(x) L and
we say that is transcendental over
From now on, R will denote an arbitrary ring with 1.
Definition. A (left) module M over R is an abelian group together with an action of R on
M (a mapping R M M ) such that for all a, b R, and all x, y M , (a + b)x = ax + bx,
a(x + y) = ax +
For us, ring will mean a ring with 1, an identity element for multiplication. There is
one exceptional ring to be careful of, namely R = cfw_0 in which 1 = 0 is both the additive
and multiplicative identity. For any ring, we denote the additiv
A Quick Trip through Category Theory
Most topics here are standard to any graduate algebra book. The concept of universal
objects for a functor is treated differently in most books and is largely ignored in general
in [S] (though it is implicit in all the
Define semigroup, monoid, group.
Definition. A function f : G H, where G, H are groups, is a group homomorphism if
f (g1 g2 ) = f (g1 )f (g2 ) for all gi G. A group homomorphism f : G H is
1) a monomorphism if it is injective;
2) an epimorphi