We have to show polynomial ring an integral domain. Whereas polynomial ring contains finite polynomials...so how can we prove it using infinite polynomials f(x) and g(x)... Aren't we supposed to differentiate it from formal power series ring??
ya ur argument is correct. On the other way to Prove that in a Ring there is no zero divisors that is we have to show that product of any two non zero elements say a and b which is not equal to zero ( a.b is not equal to zero for all a and b belongs to Ring)
It is very simple to show that D[x] is Commutative because D is integral domain Otherwise Take any 2 polynomials denote is as F(x) and g(x) take it's general form find F(x).g(x) by multiplication of polynomial u get expansion as D is integral domain u can interchange elements in products now u get g(x).f(x) this proves that D[x] is Commutative Ring
![D is an integral domain implies D[x] is integral domain](http://i.ytimg.com/vi/5KohI6mmRxs/mqdefault.jpg)
![If R is a Field then Polynomial Ring R[x] need not be a Field - Proof- ED - Lesson 24](http://i.ytimg.com/vi/9j5hhLOrkMI/mqdefault.jpg)
sir ring theory chapter ki aur theorem k proof ki video bhi upload krdo please .i will be very thankfull to you
Plz tell me the topic which u want
Very good lectures 👍🏻 keep doing
Welcome 😊!!!!
We have to show polynomial ring an integral domain. Whereas polynomial ring contains finite polynomials...so how can we prove it using infinite polynomials f(x) and g(x)... Aren't we supposed to differentiate it from formal power series ring??
Zero divisor= if a.b=0,then neither a=0,nor b=0,without zero divisor means if a.b=0 then a=0 or b=0
ya ur argument is correct. On the other way to Prove that in a Ring there is no zero divisors that is we have to show that product of any two non zero elements say a and b which is not equal to zero ( a.b is not equal to zero for all a and b belongs to Ring)
Sir plzz solve converse part of this therom
Thanks sir
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Commutative ring with unity?
It is very simple to show that D[x] is Commutative because D is integral domain
Otherwise Take any 2 polynomials denote is as F(x) and g(x) take it's general form find F(x).g(x) by multiplication of polynomial u get expansion as D is integral domain u can interchange elements in products now u get g(x).f(x) this proves that D[x] is Commutative Ring
Thannks
Thanku bro👌👌