embedding is nothing but a one-to-one homomorphism. so, for exampple, embedding a group (ring) in a field means that you may assume that is a subgroup (subring) of

for exxample, by embedding a group in a field we mean a one-to-one map such that, for all an immediate result is

that must be abelian because is abelian. similarly, embedding a ring in a field means that there exists a map such that, for all we have:

and also if has , then clearly, such has to be a (commutative) integral domian.