Example 3.0.4.

1.

Integers \(\mathbb {Z}\) form a ring with respect to the usual addition and multiplication (check the axioms)

Let \(a\, , \, b\, , \, c\in \mathbb {Z}\). Then, \(\, \mathbb {Z} = \big \{\cdots \cdots ,\, -3\, , \, -2\, , \, -1\, , \, 0\, , \, 1\, , \, 2\, , \, 3\, , \cdots \cdots \big \}\)

(i)

\(a + b = b + a\hspace {0.3cm} \forall \, a\, , \, b\in \mathbb {Z}\)

(ii)

\((a + b) + c = a + b + c = a + (b + c) \hspace {0.5cm} \forall \, a\, , \, b\, , \, c\in \mathbb {Z}\)

(iii)

\(\exists \, 0\in \mathbb {Z} \ni 0 + a = a + 0 = a\)

(iv)

\( a(bc) = abc = (ab)c \hspace {0.5cm} \forall \, a\, , \, b\, , \, c\in \mathbb {Z}\)

(v)

\(a (b+c) = ab + ac \hspace {0.5cm} \forall \, a\, , \, b\, , \, c\in \mathbb {Z}\)

(vi)

\(\exists \, -a\in \mathbb {Z} \ni (-a) + a = a + (-a) = 0\hspace {0.4cm} \forall \, a\in \mathbb {Z}\)

2.

Rational Numbers \(\mathbb {Q}\). (check the axioms).

Let \(\, a/b\, , \, c/d\, , \, e/f\in \mathbb {Q}\). Then

(i)

\(\dfrac {a}{b} + \dfrac {c}{d} = \dfrac {c}{d} + \dfrac {a}{b}\)

(ii)

\(\Big (\dfrac {a}{b} + \dfrac {c}{d}\Big ) + \dfrac {e}{f} = \dfrac {a}{b} + \dfrac {c}{d} + \dfrac {e}{f} = \dfrac {a}{b} + \Big (\dfrac {c}{d} + \dfrac {e}{f}\Big )\)

(iii)

\(\exists \, 0\in \mathbb {Q} \ni \, 0 + \dfrac {a}{b} = \dfrac {a}{b} = \dfrac {a}{b}\)

(iv)

\(\exists \, - \dfrac {a}{b} \in \mathbb {Q}\ni -\dfrac {a}{b} + \dfrac {a}{b} = \dfrac {a}{b} + \Big (-\dfrac {a}{b}\Big ) = 0\)

(v)

\(\dfrac {a}{b}\cdot \Big (\dfrac {c}{d}\cdot \dfrac {e}{f}\Big ) = \Big (\dfrac {a}{b}\cdot \dfrac {c}{d}\Big )\cdot \dfrac {e}{f}\)

(vi)

\(\dfrac {a}{b}\Big (\dfrac {c}{d} + \dfrac {e}{f}\Big ) = \dfrac {ac}{bd} + \dfrac {ae}{bf}\)

3.

Real numbers \(\mathbb {R}.\hspace {0.5cm}\) (check the axioms)

Let \(x\,, \, y\, , \, z \in \mathbb {R}\). Then

(i)

\( x + y = y + x\hspace {0.5cm} \forall \, x\, , \, y\in \mathbb {R}\)

(ii)

\( (x + y) + z = x + (y + z)\hspace {0.5cm} \forall x\, , \, y\, , \, z\in \mathbb {R}\)

(iii)

\(\exists \, 0\in \mathbb {R} \ni 0 + x = x + 0 \hspace {0.5cm}\forall x\in \mathbb {R}\)

(iv)

\(\exists \, -x \in \mathbb {R} \ni \, -x + x = x + (-x) = 0\)

(v)

\(x(yz) = (xy)z \hspace {0.5cm} \forall x\, , \, y\, , \, z\in \mathbb {R}\)

(vi)

\(x(y + z) = xy + xz \hspace {0.5cm} \forall x\, , \, y\, , \, z\in \mathbb {R}\)

4.

Even integers \(\big \{2n\, \big |\, n\in \mathbb {Z}\big \}\). \(\hspace {0.4cm}\) (Check axioms)

Let \(\, 2x\, , \, 2y\, , \, 2z\in 2n\hspace {0.3cm} \forall n\in \mathbb {Z}\). Then;

(i)

\( 2x + 2y = 2(x+ y) = 2(y+x) = 2y + 2x \hspace {0.5cm} \forall \, x\, , \, y\in \mathbb {Z}\)

(ii)

\( (2x + 2y) + 2z = 2(x + y) + 2z = 2(x + y + z) = 2x + 2y + 2z = 2x + (2y + 2z)\)

(iii)

\(\exists \, 2(0) = 0\in 2n\) for \(n = 0\ni 0 + 2x = 2x + 0 = 2x\, , \hspace {0.5cm} \forall \, 2x\in 2n\, , \, n\in \mathbb {Z}\)

(iv)

\(\exists \, -2x \) for \(-x\in \mathbb {Z} \ni -2x + 2x = 2x + (-2x) = 0\hspace {0.5cm} \forall \, 2x\in 2n\, ,\, x\in \mathbb {Z}\)

(v)

\(2x(2y\cdot 2z) = 2x\cdot 2y\cdot 2z = (2x\cdot 2y)2z. \hspace {0.5cm} \forall \, x\, , \, y\, , \, z\in \mathbb {Z}\)

(vi)

\(2x (2y + 2z) = 2x\cdot 2y + 2z\cdot 2y\hspace {0.5cm} \forall \, x\, , \, y\, , \, z\in \mathbb {Z}\)

5.

\(\mathbb {Z}_n\) where addition and multiplication are done mod\(_n\).  (Check axioms)

Let \(\, a\, , \, b\, , \, c\in \mathbb {Z}_n\). Then

(i)

\(a + b = b + a = x\) mod\(_n\) for \(x\in \mathbb {Z}_n\) or \(\, (a+b)\) mod\(_n = x = (b + a)\) mod\(_n\).

(ii)

\(\big [ (a + b) + c\big ]\) mod\(_n = y = \big [a + (b + c)\big ] \)mod\(_n\) for \(y\in \mathbb {Z}_n\)

(iii)

\(\exists \, 0 \in \mathbb {Z}_n \ni x + 0 = 0 + x = x\) where \(\, 0 = n\)mod\(_n\)

(iv)

\(\exists \, - a\in \mathbb {Z}_n \ni \, -a + a = a + (-a) = 0\), where \(\, -a + a = n\) mod\(_n\).

(v)

\(\big [a(b + c)\big ]\) mod\(_n = z = \big [(a+b) + c\big ] \)mod\(_n.\hspace {0.3cm} \forall \, a\, , \, b\, , \, c\in \mathbb {Z}_n\)

(vi)

\([a(bc)]\) mod\(_n = z = [abc] \)mod\(_n = [(ab)c] \)mod\(_n\hspace {0.5cm} \forall \, a\, , \, b\, , \, c\, , \,z\in \mathbb {Z}_n\)

6.

Let \(M(2,\mathbb {Z})\), denote the set of all \(2\times 2\) matrices over \(\mathbb {Z}\). \(\Big (\)Infact this is a unitary ring unity \(\begin {pmatrix} 1 & 0\\ 0 & 1\\ \end {pmatrix}\Big )\)

Let \(\hspace {0.1cm}\begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} = p\, , \hspace {0.2cm} \begin {pmatrix} w & x\\ y & z\\ \end {pmatrix} = q\, , \hspace {0.2cm} \begin {pmatrix} k & l\\ m & n\\ \end {pmatrix} = r\hspace {0.5cm} \forall \, a, \, b, \, c, \, d, \, k, \, l, \, m,\, n, \, x, \, y, \, z,\, w\in \mathbb {Z}\)

(i)

\(p + q = \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} + \begin {pmatrix} w & x\\ y & z\\ \end {pmatrix} = \begin {pmatrix} a + w & b + x\\ c + y & d + z\\ \end {pmatrix} = \begin {pmatrix} w + a & x + b\\ y + c & z + d\\ \end {pmatrix}\)
\[ = \begin {pmatrix} w & x\\ y & z\\ \end {pmatrix} + \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} = q + p\]

(ii)

\((p + q) + r = \begin {pmatrix} a + w & b + x\\ c + y & d + z\\ \end {pmatrix} + \begin {pmatrix} k & l\\ m & n\\ \end {pmatrix} = \begin {pmatrix} (a + w) + k & (b + x) + l\\ (c + y) + m & (d + z) + n\\ \end {pmatrix}\)
\[ = \begin {pmatrix} a + (w + k) & b + (x + l)\\ c + (y + m) & d + (z + n)\\ \end {pmatrix}\] \[ = \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} + \begin {pmatrix} w + k & x + l\\ y + m & z + n\\ \end {pmatrix} = p + (q + r)\]

(iii)

\(\exists \, \begin {pmatrix} 0 & 0\\ 0 & 0\\ \end {pmatrix} \in M(2,\mathbb {Z})\ni \begin {pmatrix} 0 & 0\\ 0 & 0\\ \end {pmatrix} + \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} = \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} + \begin {pmatrix} 0 & 0\\ 0 & 0\\ \end {pmatrix} = \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix}\hspace {0.3cm}\forall a,\, b,\, c,\, d\in \mathbb {Z}\)

(iv)

\(\exists \, -p = \begin {pmatrix} -a & -b\\ -c & -d\\ \end {pmatrix}\in M(2,\mathbb {Z})\ni \begin {pmatrix} -a & -b\\ -c & -d\\ \end {pmatrix} + \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} = \begin {pmatrix} -a + a & -b + b\\ -c + c & -d + d\\ \end {pmatrix}\) \[\begin {pmatrix} a + (-a) & b + (-b)\\ c + (-c) & d + (-d)\\ \end {pmatrix} = \begin {pmatrix} a & b\\ c & d\\ \end {pmatrix} + \begin {pmatrix} -a & -b\\ -c & -d\\ \end {pmatrix} = 0\]

(v)

(vi)

7.

Let \(\mathbb {Z}[x]\) denote all polynomials with integer coefficient with the usual addition and multiplication of polynomials. \begin {align*} \text {Let}\hspace {0.5cm} f(x) = a & = a_0 + a_1x + a_2x^2 + \cdots \cdots \cdots + a_nx^n\\ g(x) = b & = b_0 + b_1x + b_2x^2 + \cdots \cdots \cdots + b_nx^n\\ h(x) = c & = c_0 + c_1x + c_2x^2 + \cdots \cdots \cdots + c_nx^n\\ \end {align*}

(i)

\(a + b = a_0 + a_1x + a_2x^2 + \cdots \cdots \cdots + a_nx^n + b_0 + b_1x + b_2x^2 + \cdots \cdots \cdots + b_nx^n\) \begin {align*} & = a_0 + b_0 + a_1 x + b_1 x + a_2 x^2 + b_2 x^2 +\cdots \cdots \cdots + a_n x^n + b_n x^n\\ & = b_0 + a_0 + b_1 x + a_1 x + b_2 x^2 + a_2 x^2 + \cdots \cdots \cdots + b_n x^n + a_n x^n\\ & = b_0 + b_1x + b_2x^2 + \cdots \cdots \cdots + b_nx^n + a_0 + a_1x + a_2x^2 + \cdots \cdots \cdots + a_nx^n\\ & = b + a\\ \end {align*}

(ii)

\((a + b) + c = a_0 + b_0 + a_1 x + b_1 x + \cdots \cdots + a_nx^n + b_n x^n + c_0 + c_1x + c_2x^2 + \cdots \cdots \cdots + c_nx^n\) \begin {align*} & = a_0 + b_0 + c_0 + a_1 x + b_1 x + c_1 x + \cdots \cdots \cdots + a_n x^n + b_n x^n + c_n x^n\\ & = a_0 + a_1x + \cdots \cdots + a_n x^n + b_0 + c_0 + b_1 x + c_1 x + \cdots \cdots + b_n x^n + c_n x^n\\ & = a + (b + c)\\ \end {align*}

(iii)

\(\exists \,\, 0\in \mathbb {Z}[x]\ni 0 + a = 0 + a_0 + a_1x + a_2x^2 + \cdots \cdots \cdots + a_nx^n\) \begin {align*} & = a_0 + a_1x + a_2x^2 + \cdots \cdots \cdots + a_nx^n + 0\\ & = a_0 + a_1x + a_2x^2 + \cdots \cdots \cdots + a_nx^n\\ & = a\\ \end {align*}

(iv)

\(\exists \,\, -a\in \mathbb {Z}[x]\ni -a + a = -a_0 - a_1x - \cdots \cdots - a_nx^n + a_0 + a_1x + \cdots \cdots + a_nx^n\) \begin {align*} & = a_0 + a_1x + \cdots \cdots + a_nx^n - a_0 - a_1x - \cdots \cdots - a_nx^n\\ & = 0\\ & = a + (-a) = 0\\ \end {align*}

(v)

\(a(bc) = (a_0 + a_1x + \cdots \cdots + a_nx^n)\big ((b_0 + b_1x + \cdots \cdots + b_nx^n)(c_0 + c_1x + \cdots \cdots + c_nx^n)\big )\) \begin {align*} & = (a_0 + a_1x + \cdots \cdots + a_nx^n)(b_0c_0 + b_0c_1x + \cdots \cdots + b_0c_nx^n + b_1c_0x + b_1c_1x + \cdots \cdots )\\ & = a_0b_0c_0 + a_0b_0c_1 x + \cdots \cdots + a_0b_0c_nx^n + a_1xb_1c_0 + a_1xb_1c_1 + \cdots \cdots \\ & = (a_0b_0 + a_0b_1x + \cdots \cdots + a_0b_nx^n + a_1xb_0 + a_1xb_1x + \cdots )(c_0 + c_1x + \cdots \cdots + c_nx^n)\\ & = \big [(a_0 + a_1x + \cdots \cdot + a_nx^n)(b_0 + b_1x + \cdots \cdots + b_nx^n)\big ](c_0 + c_1x + \cdots \cdots + c_n x^n)\\ & = (ab)\cdot c\\ \end {align*}

(vi)

trivial

8.

Let \(\mathbb {Z}\big [\sqrt {2}\big ]\) denote the set of all numbers of the form \(a + b\sqrt {2}\, \, a\, , \, b\in \mathbb {Z}\,\) is a ring. (Check).

Let \(\, x = (a_1 + b_1\sqrt {2})\,, \, y = (a_2 + b_2\sqrt {2})\, , \, z = (a_3 + b_3\sqrt {2})\in \mathbb {Z}\sqrt {2}\hspace {0.3cm}\forall \, a_1,\, a_2,\, a_3,\, b_1,\, b_2,\, b_3\in \mathbb {Z}\), then

(i)

\(x + y = (a_1 + b_1\sqrt {2}) + (a_2 + b_2\sqrt {2}) = a_1 + a_2 + b_1\sqrt {2} + b_2\sqrt {2}\) \begin {align*} & = a_2 + a_1 + b_2\sqrt {2} + b_1\sqrt {2} = a_2 + b_2\sqrt {2} + a_1 + b_1\sqrt {2}\\ & = y + x\\ \end {align*}

(ii)

\((x + y) + z = (a_1 + a_2 + b_1\sqrt {2} + b_2\sqrt {2}) + (a_3 + b_3\sqrt {2})\) \begin {align*} & = a_1 + a_2 + a_3 + b_1\sqrt {2} + b_2\sqrt {2} + b_3\sqrt {2}\\ & = (a_1 + b_1\sqrt {2}) + (a_2 + b_2\sqrt {2} + a_3 + b_3\sqrt {2})\\ & = x + (y + z)\\ \end {align*}

(iii)

\(\exists \,\, 0 = 0 + 0\sqrt {2} \in \sqrt {2} \ni 0 + x = 0 + a_1 + b_1\sqrt {2} = a_1 + b_1\sqrt {2} + 0 \) \[ = a_1 + b_1\sqrt {2} = x\]

(iv)

\(\exists \,\, - x = -a_1 - b_1\sqrt {2}\in \mathbb {Z}\sqrt {2} \ni -x + x = -a_1 - b_1\sqrt {2}) + a_1 + b_1\sqrt {2}\) \begin {align*} & = a_1 + b_1\sqrt {2} - a_1 - b_1\sqrt {2})\\ & = x + (-x) = 0\\ \end {align*}

(v)

\(x(yz) = (a_1 + b_1\sqrt {2})\big [(a_2 + b_2\sqrt {2})(a_3 + b_3\sqrt {2})\big ] \) \begin {align*} & = (a_1 + b_1\sqrt {2})(a_2a_3 + a_2b_3\sqrt {2} + a_3b_2\sqrt {2} + 2b_2b_3)\\ & = (a_1a_2 + a_1b_2\sqrt {2} + a_2b_1\sqrt {2} + 2b_1b_2)( a_3 + b_3\sqrt {2})\\ & = \big [( a_1 + b_1\sqrt {2})( a_2 + b_2\sqrt {2})\big ]( a_3 + b_3\sqrt {2})\\ & = (xy)z\\ \end {align*}

(vi)

\(x(y + z) = (a_1 + b_1\sqrt {2})(a_2 + b_2\sqrt {2} + a_3 + b_3\sqrt {2})\) \begin {align*} & = a_1a_2 + a_1b_2\sqrt {2} + a_1a_3 + a_1b_3\sqrt {2} + a_2b_1\sqrt {2} + 2b_1b_2 + a_3b_1\sqrt {2} + 2b_1b_3\\ & = (a_1a_2 + 2b_1b_2 + a_1b_2\sqrt {2} + a_2b_1\sqrt {2}) + (a_1a_3 + 2b_1b_3 + a_1b_3\sqrt {2} + a_3b_1\sqrt {2})\\ & = xy + xz\\ \end {align*}

9.

Let \(R\) and \(S\) be rings, define \(\, R\times S = \big \{(r,s)\, \big |\, r\in R\, , \, s\in S\big \}, \) with addition
\((r_1\, , \, s_1) + (r_2\, ,\,s_2) = (r_1 + r_2\,, \, s_1 + s_2)\, \) and multiplication \(\, (r_1\, ,\, s_1)(r_2\, ,\, s_2) = (r_1r_2)(s_1s_2)\,\) is a ring. (Check with the axioms)