Weisstein, Eric W.
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 .
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