## Chapter 9

## Rings

#### 9.1 Ring

**Definition 9.1.** *A non-empty set R equipped with two binary operations + and (usually called addition and multiplication) is called a ring if*

- (
*R,**+*) is an Abelian group, *(R,.) is a semigroup,*- .
*is right as well as left distributive over*+,*that is**a*.(*b*+*c*) =*a.b*+*a.c*∀*a*,*b*,*c*∈*R*(

*a*+*b*).*c*=*a.c*+*b.c*∀*a*,*b*,*c*∈*R**It is denoted by (*.*R,*+, ·)

When the operations are understood we simply say that R is a ring. Moreover we use juxtaposition instead of •. Since *(R,* +) is an Abelian group, therefore

- the addditive identity is unique,
- additive inverse of an element is unique,
- cancellation laws hold for addition.

The additive identity of a ring is called the zero element and is denoted by 0.

This should not be confused ...

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