A unit ring is a ring with a multiplicative identity. It is therefore sometimes also known as a "ring with identity."
It is given by a set together with two binary operators satisfying the following conditions:
1. Additive associativity: For all , ,
2. Additive commutativity: For all , ,
3. Additive identity: There exists an element such that for all ,
4. Additive inverse: For every , there exists a such that ,
5. Multiplicative associativity: For all , ,
6. Multiplicative identity: There exists an element such that for all , ,
7. Left and right distributivity: For all , and .