Can there be more ideals than elements of a ring? Can there be more ideals than elements of a ring? This is related to my other question https://math.stackexchange.com/questions/713610/having-elements-as-ideals . At first glance, it seems obvious that there would be less ideals than elements of a ring. But the ideals are not necessarily mutually disjoint, and they form a subset of the power set of the ring, which is strictly larger than the ring itself. I know fields only have trivial ideals, so maybe there is an "opposite" structure that has a maximal proliferation of ideals relative to elements? A (weaker) related question is if there can be more subrings of than elements of a ring. 

Start with the polynomial ring, over your favorite countable field $K$, generated by countably many indeterminates $X_q$ indexed by the rational numbers $q$.  Let $R$ be the quotient of this ring obtained by imposing the equations $X_qX_r=X_q$ for all $q<r$.  (In fancier language, this is the semigroup algebra over $K$ of the semigroup consisting of the rational numbers with the operation $\min$.) Then $R$ is countably infinite, but every real number $\alpha$ gives rise to an ideal $I_\alpha$ generated by all the $X_q$ with $q<\alpha$.  So this countable ring has continuum many ideals.
Actually, it wasn't necessary to pass from the polynomial ring $K[X_q:q\in\mathbb Q]$ to the quotient ring $R$.  $K[X_q:q\in\mathbb Q]$ itself has continuum many ideals, namely the pre-images of the $I_\alpha$'s (and lots more besides). 
