Question

Show that the set z of all integers is a ring with respect addition and multiplication of integers as the two ring composition.

Answer 1

The figures are based on GDP (Nominal) and sector composition ratios provided by the CIA World Fact Book. Agriculture includes farming, fishing, and forestry. Industry includes mining, manufacturing, energy production, and construction. Services cover government activities, communications, transportation, finance, and all other private economic activities that do not produce material goods.

Agriculture Sector : Agriculture Sector contributes 6.4 percent of total world's economic production. Total production of sector is $5,084,800 million. China is the largest contributer followed by India. China and India accounts for 19.49 and 7.39 percent of total global agricultural output. World's largest economy United States is at third place. Next in line come Brazil and Indonesia.

Out of 226 countries, In nine countries, Agriculture sector is the dominant sector of their economy. Sierra Leone has 60.7% of their GDP from the agricultural sector. Three countries receive more than 50% of their GDP from the agricultural sector. Gibraltar, Macau, Monaco and Singapore has no agricultural output. World's largest countries United States has only 0.9% of GDP in Agriculture Sector.

Industry Sector : With GDP of $23,835 billion, Industry Sector holds a share of 30% of total GDP nominal. China is the largest contributer followed by US. Japan is at 3rd and Germany is at 4th place. These four countries contributes 45.84 of total global industrial output.

Industry Sector is the leading sector of 15 countries. Libya has highest share of 63.8% of their GDP. 10 countries has more than 50% of their GDP from Industry.

Services Sector : Services sector is the largest sector of the world as 63 percent of total global wealth comes from services sector. United States is the largest producer of services sector with around 15.53 trillion USD. Services sector is the leading sector in 201 countries/economies. 30 countries receive more than 80 percent of their GDP from services sector. Chad has lowest 27% contribution by services sector in its economy.

Gibraltar has 100% of gdp in services sector, other two sectors has zero output.

Answer 2

Given:

The set of integers is,

$\[Z = \left\{ {....... - 3,\, - 2,\, - 1,\,0,\,1,\,2,\,3........} \right\}\]$

The binary operations are addition and multiplication denoted by $\[\left( + \right)\]$ and $\[\left( . \right)\]$ respectively.

To Prove:

The set $Z$ of all integers is a ring with respect to addition and multiplication as the composition i.e., $\[\left( {Z,\, + ,\,.} \right)\]$ is a ring.

Solution:

To prove that $\[\left( {Z,\, + ,\,.} \right)\]$ is a ring, we must show that the following axioms are satisfied:

1. $\[\left( {Z,\, +} \right)\]$ is an abelian group.
2. $\[\left( {Z,\,.} \right)\]$ is a semi group.
3. Multiplication is distributive over addition for all the integers.

1. First to prove $\[\left( {Z,\, +} \right)\]$ is an abelian group:

(i) The sum of two integers is again an integer. Thus, $Z$ is closed under addition i.e., $\[a,\,b \in Z \Rightarrow a + b \in Z\]$

(ii) Associative law holds for all integers under addition i.e., $\[\left( {a + b} \right) + c = a + \left( {b + c} \right)\,\forall \,a,\,b,\,c \in Z\]$

(iii) $0$ is the identity element for all integers i.e., $\[0 + a = a + 0 = a\,\forall \,a \in Z\]$

(iv) Inverse of an integer $a$ is $-a$ i.e., $\[a + \left( { - a} \right) = 0\,\forall \,a,\, - a \in Z\]$

(v) All integers follow commutativity under addition i.e., $\[a + b = b + a\,\forall \,a,\,b \in Z\]$

Hence, $\[\left( {Z,\, +} \right)\]$ is an abelian group.

2. Now to prove $\[\left( {Z,\,.} \right)\]$ is a semi group:

(i) The multiplication of two integers is again an integer. Thus, $Z$ is closed under multiplication i.e., $\[a,\,b \in Z \Rightarrow a . b \in Z\]$

(ii) Integers obey associative law for multiplication i.e., $\[\left( {a . b} \right) . c = a . \left( {b . c} \right)\,\forall \,a,\,b,\,c \in Z\]$

Hence, $\[\left( {Z,\,.} \right)\]$ is a semi group.

3. Multiplication is distributive over addition for all the integers:

Since $\[a.\left( {b + c} \right) = a.b + a.c\]$ and $\[\left( {a + b} \right).c = a.c + b.c\,\,\forall \,a,\,b,\,c \in Z\]$, therefore multiplication is distributive over addition for all the integers on both the left and the right.

Therefore, satisfying all the above three axioms 1., 2. and 3., the set  $\[\left( {Z,\, + ,\,.} \right)\]$ is a ring.

Hence, proved that the set $Z$ of all integers is a ring with respect to addition and multiplication as the composition.

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