Those sources are crazy. The ideals form a lattice under intersection and "+". All of algebraic geometry (a very popular branch of mathematics for studying rin

Those sources are crazy. The ideals form a lattice under intersection and "+". All of algebraic geometry (a very popular branch of mathematics for studying rings) is built on the lattice of ideals. There is no way of giving a ring this structure without the ring itself being the top element of this lattice. What you probably mean to say is that there are sources that do not treat R as a prime ideal of itself.