There are in fact quite a few mathematical definitions for simple. In addition to what Noether writes,

• a smooth curve curve is said to be simple iff it does not intersect itself at any point (well, possibly the endpoint if it is also closed).
• a representation is simple, or irreducible, iff it has no proper non-trivial invariant subspaces.
• a path connected topological space is said to be simply connected iff its fundamental group is trivial.
• a tensor is simple if it can be expressed as a product of vectors, eg A=aa^T, Q^ab_c^d=p^aq^br_cp^d