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Algebra-II: Companion notes

Abhay Soman

Section 1.2 Examples of Rings

We give several examples of rings with unity.
A trivial group \(\{0\}\) is a ring and is called a trivial ring. Note that in a trivial ring \(1=0\text{.}\) In fact, if \(1=0\) in a ring then that ring is trivial.
The set of integers \(\Z\) is a ring with the usual addition and the multiplication.
The set of complex numbers \(\C\text{,}\) the real numbers \(\R\text{,}\) and rationals \(\Q\) are rings with the usual addition and the multiplication.
Let \(i=\sqrt{-1}\in\C\text{.}\)The subset of \(\C\)
\begin{equation*} \Z[i]=\{a+ib:a,b\in\Z\} \end{equation*}
is a ring under the addition and the multiplication inherited from \(\C\text{.}\) This ring is called the ring of Gaussian integers. Given any \(x\in\Z[i]\) there are unique \(a,b\in\Z\) such that \(z=a+ib\text{.}\)
Consider a cyclic group \(\Z/n\Z\text{.}\) We write the coset of an integer \(i\) in \(\Z/n\Z\) as \([i]\text{.}\) We define the multiplication as follows.
\begin{equation*} [i]\cdot[j]=[ij] \end{equation*}
Here \(ij\) is the multiplication of integers \(i,j\) in \(\Z\text{.}\)
Let \(R\) be a ring with unity. Consider the set of \(n\times n\) matrices over \(R\text{,}\) \(M_n(R)\text{.}\) So
\begin{equation*} M_n(R)=\left\{(a_{ij})=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{pmatrix}:a_{ij}\in R\right\}. \end{equation*}
It is an abelian group under matrix addition.
\begin{equation*} (a_{ij})+(b_{ij})=(a_{ij}+b_{ij}) \end{equation*}
The matrix multiplication gives \(M_{n}(R)\) a monoid structure.
\begin{equation*} \begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{pmatrix}\cdot\begin{pmatrix}b_{11}\amp b_{12}\amp\cdots\amp b_{1n}\\b_{21}\amp b_{22}\amp\cdots\amp b_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\b_{n1}\amp b_{n2}\amp\cdots\amp b_{nn}\end{pmatrix}=\begin{pmatrix}\sum_ja_{1j}b_{j1}\amp \sum_ja_{1j}b_{j2}\amp\cdots\amp \sum_ja_{1j}b_{jn}\\\sum_ja_{2j}b_{j1}\amp \sum_ja_{2j}b_{j2}\amp\cdots\amp \sum_ja_{2j}b_{jn}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\\sum_ja_{nj}b_{j1}\amp \sum_ja_{nj}b_{j2}\amp\cdots\amp \sum_ja_{nj}b_{jn}\end{pmatrix} \end{equation*}
Consider an abelian group \(A\) with the group operation written additively. We write the additive identities of \({\mathbb{Z}}\) and \(A\) by \(0_{\mathbb{Z}}\) and \(0_A\text{,}\) respectively. For \(n\in\mathbb{N}\) and \(a\in A\) we make following conventions.
Left action:
\begin{equation} n\cdot a=\underbrace{a+a+\cdots+a}_{n-\text{times}}\quad\text{and}\quad -n\cdot a=\underbrace{(-a)+(-a)+\cdots+(-a)}_{n-\text{times}}\tag{1.2.1} \end{equation}
Right action:
\begin{equation} a\cdot n=\underbrace{a+a+\cdots+a}_{n-\text{times}}\quad\text{and}\quad a\cdot (-n)=\underbrace{(-a)+(-a)+\cdots+(-a)}_{n-\text{times}}\tag{1.2.2} \end{equation}
For \(0_{\mathbb{Z}}\in\mathbb{Z}\) and any \(a\in A\) we take
\begin{equation} 0_{\mathbb{Z}}\cdot a=0_A\quad\text{and}\quad a\cdot0_{\mathbb{Z}}=0_A.\tag{1.2.3} \end{equation}
Consider a triangular ring.
\begin{equation*} R=\left\{\begin{pmatrix}m\amp a\\0_{\mathbb{Z}}\amp n\end{pmatrix}:m,n\in\mathbb{Z}\text{ and }a\in A\right\} \end{equation*}
The addition on \(R\) is defined as follows.
\begin{equation*} \begin{pmatrix}m_1\amp a_1\\0_{\mathbb{Z}}\amp n_1\end{pmatrix}+\begin{pmatrix}m_2\amp a_2\\0_{\mathbb{Z}}\amp n_2\end{pmatrix}=\begin{pmatrix}m_1+m_2\amp a_1+a_2\\0_{\mathbb{Z}}\amp n_1+n_2\end{pmatrix} \end{equation*}
The multiplication on \(R\) is defined as follows.
\begin{equation} \begin{pmatrix}m_1\amp a_1\\0_{\mathbb{Z}}\amp n_1\end{pmatrix}\cdot\begin{pmatrix}m_2\amp a_2\\0_{\mathbb{Z}}\amp n_2\end{pmatrix}=\begin{pmatrix}m_1 m_2\amp m_1\cdot a_2+a_1\cdot n_2\\0_{\mathbb{Z}}\amp n_1n_2\end{pmatrix}\tag{1.2.4} \end{equation}
This is the usual matrix multiplication when we take into consideration the left and the right actions defined in (1.2.1) and (1.2.2), respectively.
Let \(R\) be a ring. Consider the polynomial ring in one variable over \(R\text{:}\)
\begin{equation*} R[X]=\left\{r_0+r_1X+\cdots +r_nX^n:r_i\in R\text{ and }n\in\mathbb{N}\right\}. \end{equation*}
Addition and multiplication is defined in usual way:
\begin{gather*} \text{Addition:}\;\text{ if }m\leq n\text{ then}\quad\sum_{i=0}^mr_iX^i+\sum_{j=0}^ns_jX^j=\sum_{k=0}^{m}(r_k+s_k)X^k+\sum_{j=m+1}^{n}s_jX^j\\ \text{We also assume }\sum_ia_iX^i+\sum_jb_jX^j=\sum_jb_jX^j+\sum_ia_iX^i\;\text{for any two elements of }R[X]\\ \text{Multiplication:}\quad\left(\sum_{i=0}^{m}r_iX^i\right)\left(\sum_{j=0}^{n}s_jX^j\right)=\sum_{i=0}^{m}\left(\sum_{j=0}^{n}r_is_jX^{j+i}\right)=r_0s_0+\sum_{i=1}^{m+n}\left(\sum_{j=0}^{i}r_js_{i-j}\right)X^i\\ \text{We also assume that }\sum_ir_iX^i=\sum_is_iX^i\;\text{if and only if}\;r_i=s_i\;\text{for every }i. \end{gather*}
Let \(R\) be a ring. The opposite ring of \(R\) is denoted by \(R^{\rm op}\) and is defined by the following properties.
  1. We denote elements of \(R^{\rm op}\) by \(r^{\rm op}\text{.}\) Thus when \(r\in R\) is viewed as an element of \(R^{\rm op}\) we denote it by \(r^{\rm op}\text{.}\)
  2. The underlying set and the abelian group structure of \(R^{\rm op}\) is the same as that of \(R\text{.}\) In other words, \((R^{\rm op},+)=(R,+)\text{.}\) Hence for \(r,s\in R\) we have
    \begin{equation*} r^{\rm op}+s^{\rm op}=(r+s)^{\rm op}. \end{equation*}
  3. The multiplication is defined as follows: for \(r^{\rm op},s^{\rm op}\in R^{\rm op}\text{,}\) i.e., for \(r,s\in R\)
    \begin{equation*} r^{\rm op}s^{\rm op}=(sr)^{\rm op}. \end{equation*}
If for every \(r,s\in R\text{,}\) \(rs=sr\) then the multiplication in \(R^{\rm op}\) is also the same as that of \(R\text{.}\)
Let \(I\) be a nonempty indexing set and for each \(i\in I\) let \(R_i\) be a ring. The product ring \(\prod_iR_i\) is a ring with componentwise addition and multiplication.
\begin{gather*} (\ldots,r_i,\ldots)+(\ldots,s_i,\ldots)=(\ldots,r_i+s_i,\ldots)\\ (\ldots,r_i,\ldots)(\ldots,s_i,\ldots)=(\ldots,r_is_i,\ldots) \end{gather*}
Similar to the above construction we have the following example.
Let \(R\) be a ring and let \(I\) be a nonempty set. Let \(R^I\) be the set of all functions from \(I\) to \(R\text{.}\) The set \(R^I\) is a ring with following operations of addition and multiplication.
\begin{gather*} \text{Addition:}\quad (f+g)(i)=f(i)+g(i)\\ \text{Additive Identity:}\quad\mathbf{0}\colon I\to R\quad\text{give by}\quad i\mapsto 0\quad\text{for every}\;i\in I\\ \text{Multiplication:}\quad (fg)(i)=f(i)g(i)\\ \text{Multiplicative Identity:}\quad \unit\colon I\to R\quad\text{given by}\quad i\mapsto 1\quad\text{for every}\quad i\in I \end{gather*}
This example is related to Example 1.2.6 (recall from Linear algebra course). Let \(V\) be a vector space over a field \(F\text{.}\) Consider the set of all endomorphisms of \(V\text{,}\) \(\End_F{V}\) i.e., the set of all \(F\)-linear maps from \(V\) to \(V\text{.}\) This is a ring under following operations.
  1. Addition: \((f+g)(v)=\underbrace{f(v)+g(v)}_{\text{addition in }V}\) for \(f,g\in\End_F(V)\) and \(v\in V\)
  2. Multiplication: \((f\circ g)(v)=f\left(g(v)\right)\) for \(f,g\in\End_F(V)\) and \(v\in V\)
Let \(I\) be a nonempty open interval of \(\R\text{.}\) The set of all continuous real-valued functions from \(I\) to \(\R\text{,}\)
\begin{equation*} C(I)=\{f\colon I\to\R:f\text{ is continuous}\} \end{equation*}
is a ring under addition of functions and the multiplication defined as follows.
\begin{equation*} \left(f\cdot g\right)(x)=f(x)g(x)\quad\text{for}\; f,g\in C(I)\;\text{and}\;x\in I. \end{equation*}
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