  Two players alternate moves in the following impartial combinatorial game:
Given a finitely generated abelian group $A$, a move consists of picking some
nonzero element $a \in A$. The game then continues with the quotient group $A/
\langle a \rangle$. We prove that under the normal play rule, the second player
has a winning strategy if and only if $A$ is a square, i.e. $A$ is isomorphic
to $B \times B$ for some abelian group $B$. Under the mis\`ere play rule, only
minor modifications concerning elementary abelian groups are necessary to
describe the winning situations. We also compute the nimbers, i.e.
Sprague-Grundy values, of $2$-generated abelian groups. An analogous game can
be played with arbitrary algebraic structures. We study some examples of
non-abelian groups and commutative rings such as $R[X]$, where $R$ is a
principal ideal domain.
