Section 1.1 Definition
Definition 1.1.1. (Ring with unity).
A ring \(R\) with unity is a nonempty set with two binary operations addition, denoted by ‘\(+\)’ and multiplication, denoted by ‘\(\cdot\)’ satisfying following conditions.
-
[Group under addition].
\((R,+)\) is an abelian group. We denote by \(0\) the identity of this group.
-
[Associativity under multiplication].
For any \(r,s,t\in R\) we have \(r\cdot(s\cdot t)=(r\cdot s)\cdot t\text{.}\)
-
[Existence of Unity].
There is an element
\(1\in R\text{,}\) called
the unity of \(R\), such that for every
\(r\in R\) we have
\begin{equation*}
r\cdot 1=1\cdot r=r.
\end{equation*}
-
[Distributive Laws].
For any
\(r,s,t\in R\) we have
\begin{equation*}
r\cdot(s+t)=r\cdot s+r\cdot t\quad\text{and}\quad(s+t)\cdot r=s\cdot r+t\cdot r.
\end{equation*}
We give examples of rings in the next sections. We first deduce some properties of rings with unity.
Proposition 1.1.6.
Let \(R\) be a ring with unity. We have the following properties.
\(\displaystyle \left(\sum_{i=1}^{m}r_i\right)\cdot\left(\sum_{j=1}^{n}s_j\right)=\sum_{i,j}(r_i\cdot s_j)\)
For any
\(r,s\in R\) we have
\begin{equation*}
0\cdot r=r\cdot 0=0\quad\text{and}\quad (-r)\cdot s=-(rs)\text{,}
\end{equation*}
where
\(-r\) and
\(-(rs)\) are the additive inverses of
\(r\) and
\(rs\text{,}\) respectively. In particular, the additive inverse of
\(r\in R\) is
\(-1\cdot r\text{.}\)
If
\(r\in R\) commutes with
\(s\in R\text{,}\) i.e., if
\(r\cdot s=s\cdot r\) then we have the
binomial theorem.
\begin{equation*}
(r+s)^n=r^n+{n\choose 1} r^{n-1}\cdot s+\cdots+s^n
\end{equation*}
