**Definition.** *By a* ring *we mean a nonempty set R with two binary operations + and ·, called addition and multiplication (also called product), respectively, such that*

*(i)* (*R,+*) *is an additive abelian group.*

*(ii)* (*R, ·*) *is a multiplicative semigroup.*

*(iii) Multiplication is distributive (on both sides) over addition; that is, for all* ,

*(The two distributive laws are respectively called the left distributive law and the right distributive law.)*

We usually write *ab* instead of *a · b*. The identity of the ...

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