I'll mention some more useful non-examples:
A non-monoid semigroup: $({\mathbb Z^+},+)$.
A non-group monoid: $({\mathbb N}, +)$, $({\mathbb Z},\cdot)$.
A non-integral domain: ${\mathbb Z}_6$.
A noetherian non-artinian ring: the integers ${\mathbb Z}$, the ring of Laurent polynomials over a field $K[x,x^{-1}]$.
A non-unital semisimple ring: the row-finite, column-finite, infinite matrices over a field ${\mathbb M}_{\infty}(K)$.
A simple non-semisimple algebra: The Weyl algebra $K[X,Y]/(XY-YX-1)$.
A non-simple indecomposable ring: $K[x,x^{-1}]$.
