Examples of Rings
Example 1:
A Gaussian integer is a complex number $$a + ib$$, where $$a$$ and $$b$$ are integers. Show that the set $$J\left( i \right)$$ of Gaussian integers forms a ring under the ordinary addition and multiplication of complex numbers.
Solution:
Let $${a_1} + i{b_1}$$ and $${a_2} + i{b_2}$$ be any two elements of $$J\left( i \right)$$, then
\[\left( {{a_1} + i{b_1}} \right) + \left( {{a_2} + i{b_2}} \right) = \left( {{a_1} + {a_2}} \right) = i\left( {{b_1} + {b_2}} \right) = A + iB\] and
\[\left( {{a_1} + i{b_1}} \right) \cdot \left( {{a_2} + i{b_2}} \right) = \left( {{a_1}{a_2} – {b_1}{b_2}} \right) + i\left( {{a_1}{b_2} + {b_1}{a_2}} \right) = C + iD\]
These are Gaussian integers and therefore $$J\left( i \right)$$ is closed under addition as well as the multiplication of complex numbers. Addition and multiplication are both associative and commutative compositions for complex numbers.
Also, multiplication distribution with respect to addition. The additive inverse of $$a + ib \in J\left( i \right)$$ is $$\left( { – a} \right) + \left( { – b} \right)i \in J\left( i \right)$$ as
\[\begin{gathered} \left( {a + ib} \right) = \left( { – a} \right) + \left( { – b} \right)i \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {a – a} \right) + \left( {b – b} \right)i \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 0 + 0i = 0 \\ \end{gathered} \]
The Gaussian integer $$1 + 0 \cdot i$$ is the multiplicative identity. Therefore, the set of Gaussian integers is a commutative ring with unity.
Example 2: Prove that the set of residue {0, 1, 2, 3, 4} modulo 5 is a ring with respect to the addition and multiplication of residue classes (mod 5).
Solution: Let R = {0, 1, 2, 3, 4}. Addition and multiplication tables for given set R are:
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+ mod 5
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0
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1
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2
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3
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4
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mod 5
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0
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1
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2
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3
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4
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0
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0
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1
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2
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3
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4
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0
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0
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0
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0
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0
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0
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1
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1
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2
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3
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4
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0
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1
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0
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1
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2
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3
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4
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2
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2
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3
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4
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0
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1
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2
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0
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2
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4
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1
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3
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3
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3
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4
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0
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1
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2
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3
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0
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4
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4
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0
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1
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2
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3
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4
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From the addition composition table the following is clear:
(i) Since all elements of the table belong to the set, it is closed under addition (mod 5).
(ii) Addition (mod 5) is always associative.
(iii) $$0 \in R$$ is the identity of addition.
(iv) The additive inverse of the elements 0, 1, 2, 3, 4 are 0, 4, 3, 2, 1 respectively.
(v) Since the elements equidistant from the principal diagonal are equal to each other, the addition (mod 5) is commutative.
From the multiplication composition table, we see that (R, .) is a semi group, i.e. following axioms hold good.
(vi) Since all the elements of the table are in R, the set R is closed under multiplication (mod 5).
(vii) Multiplication (mod 5) is always associative.
(viii) The multiplication (mod 5) is left as well as right distributive over addition (mod 5).
Hence $$\left( {R, + , \cdot } \right)$$ is a ring.