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If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\Bbb R[x,y]$ modulo the ideal $(x^2+y^2-1)$. Then $x^2=(1-y)(1+y)$ and likewise $y^2=(1-x)(1+x)$. (Over the complex numbers we also have $1=(x+iy)(x-iy)$, and, as Georges points out, the quotient ring is in fact a UFD.)

My question is: are these (in some sense) the only examples which can be factored in different ways? Let me explain what I mean: The above quotient ring (over the reals), call it $A$, is Noetherian so every element can be factored into irreducible ones. I'd be interested to see further elements (not a multiple of the above ones) which don't factorize uniquely. Can something interesting be said about those elements?

 

What happens in more general quotient rings (let's assume they are domains)?

Thanks for any help and pointers (in particular, the ones already received!), as well as for your patience if this is trivial.

If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\Bbb R[x,y]$ modulo the ideal $(x^2+y^2-1)$. Then $x^2=(1-y)(1+y)$ and likewise $y^2=(1-x)(1+x)$. (Over the complex numbers we also have $1=(x+iy)(x-iy)$, and, as Georges points out, the quotient ring is in fact a UFD.)

My question is: are these (in some sense) the only examples which can be factored in different ways? Let me explain what I mean: The above quotient ring (over the reals), call it $A$, is Noetherian so every element can be factored into irreducible ones. I'd be interested to see further elements (not a multiple of the above ones) which don't factorize uniquely. Can something interesting be said about those elements?

What happens in more general quotient rings (let's assume they are domains)?

Thanks for any help and pointers (in particular, the ones already received!), as well as for your patience if this is trivial.

If K is a field then the polynomial ring K[x₁,..,x_n] is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, R[x,y] modulo the ideal (x²+y²-1). Then x²=(1-y)(1+y) and likewise y²=(1-x)(1+x). (Over the complex numbers we also have 1=(x+i y)(x-i y), and, as Georges points out, the quotient ring is in fact a UFD.)

My question is: are these (in some sense) the only examples which can be factored in different ways? Let me explain what I mean: The above quotient ring (over the reals), call it A, is Noetherian so every element can be factored into irreducible ones. I'd be interested to see further elements (not a multiple of the above ones) which don't factorize uniquely. Can something interesting be said about those elements?

What happens in more general quotient rings (let's assume they are domains)?

Thanks for any help and pointers (in particular, the ones already received!), as well as for your patience if this is trivial.

If $K$ is a field then the polynomial ring $K[x_1,\ldots, x_n]$ is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, $\Bbb R[x,y]$ modulo the ideal $(x^2+y^2-1)$. Then $x^2=(1-y)(1+y)$ and likewise $y^2=(1-x)(1+x)$. (Over the complex numbers we also have $1=(x+iy)(x-iy)$, and, as Georges points out, the quotient ring is in fact a UFD.)

My question is: are these (in some sense) the only examples which can be factored in different ways? Let me explain what I mean: The above quotient ring (over the reals), call it $A$, is Noetherian so every element can be factored into irreducible ones. I'd be interested to see further elements (not a multiple of the above ones) which don't factorize uniquely. Can something interesting be said about those elements?

What happens in more general quotient rings (let's assume they are domains)?

Thanks for any help and pointers (in particular, the ones already received!), as well as for your patience if this is trivial.

If K is a field then the polynomial ring K[x₁,..,x_n] is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, R[x,y] modulo the ideal (x²+y²-1). Then x²=(1-y)(1+y) and likewise y²=(1-x)(1+x). (Over the complex numbers we also have 1=(x+i y)(x-i y) which makes the involved factors units.)

My question is: are these (in some sense) the only examples which can be factored in different ways? If there are more (fundamentally different, eg. not just multiples) examples, can one classify them?

In more generality: is it true and is there a constructive way to make sense of quotients of polynomial rings as "almost UFDs"? (Constructive means: given generators for the ideal, how can we, if possible, find a list of the different factorizations.)

Thanks for any help and pointers, as well as for your patience if this is trivial.

If K is a field then the polynomial ring K[x₁,..,x_n] is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, R[x,y] modulo the ideal (x²+y²-1). Then x²=(1-y)(1+y) and likewise y²=(1-x)(1+x). (Over the complex numbers we also have 1=(x+i y)(x-i y), and, as Georges points out, the quotient ring is in fact a UFD.)

My question is: are these (in some sense) the only examples which can be factored in different ways? Let me explain what I mean: The above quotient ring (over the reals), call it A, is Noetherian so every element can be factored into irreducible ones. I'd be interested to see further elements (not a multiple of the above ones) which don't factorize uniquely. Can something interesting be said about those elements?

What happens in more general quotient rings (let's assume they are domains)?

Thanks for any help and pointers (in particular, the ones already received!), as well as for your patience if this is trivial.