3 One more example

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: $K[x,x^{-1}]$.

I'll mention some more useful non-examples:

A non-monoid semigroup: $({\mathbb Z^+},+)$.

A non-group monoid: $({\mathbb Z},\cdot)$.

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: $K[x,x^{-1}]$.

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: $K[x,x^{-1}]$.