Question

How many property contain a ring

Answer 1

For the first one occurring, you could use the ideal $(x)/(x^3)$ in $F_2[x]/(x^3)$. It is properly inside a local ring, so it doesn't have any idempotents aside from $0$, so no identity is there. It's not a domain since $x^3=0$.

For the second one, the integer quaternions (the subring of the quaternions generated by $i,j,k$ and the integers) (or also the Hurwitz quaternions) will do. They're a domain with identity that isn't commutative.

For the third one, you can take any nontrivial ideal of the integer quaternions.

Answer 2

Answer:

A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms. R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).