Question

सिद्ध कीजिये कि एक परिमित, अशून्य भाजक रहित क्रमविनिमेय वलय एक क्षेत्र होता है।
Prove that a finite, non-zero commutative rings without zero division is a field.​

Answer 1

Answer:

First, we show that R has an identity. (Sometimes the existence of an identity is included in the definition of a ring, but you do not need it here.)

For each 0≠a∈R, consider the map

ϕa:R→R,x↦ax.

Because a is not a zero-divisor, the map ϕa is injective, thus surjective. So there is e∈R such that

a=ϕa(e)=ea.(1)

Again because the map ϕa is surjective, every b∈R is of the form

b=ϕa(xb)=axb

for some xb, so multiplying (1) by xb you find

b=eb

for all b∈R, so e is the identity.

Now use once more the fact that ϕa is surjective, to show that there is c∈R such that

ac=ϕa(c)=e,

so c is an inverse of a, a being an arbitrary non-zero element.

Step-by-step explanation:

Hope it Helps.