Examples of properties not preserved under homomorphism An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the second.
However, homomorphisms only indicate that the two structures are "similar", so it's not quite as obvious that every property will be preserved. Yet all the properties I've ever seen are preserved under homomorphism: commutativity, cyclicality, solvability...
What are some examples of properties of algebraic structures not preserved under homomorphism? Feel free to use any algebraic structures you like, but I'm particularly interested in your garden variety structures: group and rings, say.

An image of an algebraic object is equivalently a quotient in the most elementary cases. Taking a quotient is an identification process, so a general class of properties not preserved under images are those relating to uniqueness of solutions of equations. 
For instance, in any free abelian group a linear equation with a solution has only one solution-but in abelian groups with torsion there may be many. Similarly, rings of polynomials over a field admit factorization theorems to the effect that a polynomial of degree $n$ has no more than $n$ roots, whereas there are nonzero polynomials over finite rings that annihilate the entire ring. 
