Ring

Let $R$ be a nonempty set endowed with $2$ closed binary operations $+$and $\cdot$

$(R, +, \cdot)$ is a ring if the following conditions hold for all $a, b, c, \in R$.

• $a + b = b + a$
• $a + (b + c) = (a + b) + c$
• There exists $z \in R$ such that $a + z = z + a = a$ for every $a \in R$ ( $z$ is additive identity, must exist in a ring)
• For each $a \in R$, there is a $b \in R$ with $a + b = b + a = z$
• $a \cdot (b \cdot c) = (a \cdot b) \cdot c$
• $a \cdot (b + c) = a \cdot b + a \cdot c$ and $(b + c ) \cdot a = b \cdot a + c \cdot a$ for all $a , b, c, \in R$

In addition, the ring is said to be commutative if $a \cdot b = b \cdot a$ for all $a, b \in R$

A $u \in R$ is called a multiplicative identity or unity if $u \neq z$ and $a \cdot u = u \cdot a = a$ for all $a \in R$

• may not exist in a ring.
• if a ring contains it, then it is called a ring with unity

An element $b \in R$ is said to be $a$’s multiplicative inverse if

$$ a \cdot b = b \cdot a = 1 $$

• not guaranteed to exist in a ring
• if $a \in R$ has a multiplicative inverse, it is called a unit.

Some basic facts

$(\mathbb{Z}, +, \cdot), (\mathbb{Q}, +, \cdot), (\mathbb{R}, +, \cdot), (\mathbb{C}, +, \cdot)$ are all rings.

• The additive identity is 0.