1) (Nagata) There are noetherian domains of infinte Krull dimension: Localize $k[x_1,x_2,...]$ at the prime ideals $(x_1),(x_2,x_3),(x_4,x_5,x_6),...$.
2) (Malcev) Every commutative cancellative monoid embeds into a group. This is false in the non-commutative case. A very instructive counterexample is given by $\langle a,b,c,d,x,y,u,v : ax=by, cx=dy, au=bv \rangle$.
3) The Theorem of Cantor-Bernstein for sets does not carry over to algebraic structures. For example, the fields $K=\overline{\mathbb{Q}(x_1,x_2,...)}$ (or $K=\mathbb{C}$) and $K(t)$ embed into each other, but they are not isomorphic.