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edit approved Sep 17, 2020 at 22:17
user26857
edit approved Sep 17, 2020 at 22:17
user26857
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No,quotients quotients of polynomial rings are definitely not "almost UFDs".

Any finitely generated ring over K$K$ is such a quotient and this means a lot of non UFDs. Said differently, any algebraic variety in affine space over K$K$ has as ring of regular functions one of your quotients and in general (as your own example over $\mathbb R$ states) it will not be a UFD.

By the way, your parenthetical remark about the complex case is a bit ambiguous:  $\mathbb C[x,y]/(x^2+y^2-1)$ IS a UFD: by the change of variables u=x+iy,v=x-iy$u=x+iy,\ v=x-iy$ this ring becomes $\mathbb C[u,v]/(uv-1)=\mathbb C[u,1/u]$, which is factorial and even a PID. ( ReasonReason: If $A$ is a PID, so is every ring of fractions $A_S$ [...unless it is a field, you Bourbakistas].)

No,quotients of polynomial rings are definitely not "almost UFDs".

Any finitely generated ring over K is such a quotient and this means a lot of non UFDs. Said differently, any algebraic variety in affine space over K has as ring of regular functions one of your quotients and in general (as your own example over $\mathbb R$ states) it will not be a UFD.

By the way, your parenthetical remark about the complex case is a bit ambiguous:  $\mathbb C[x,y]/(x^2+y^2-1)$ IS a UFD: by the change of variables u=x+iy,v=x-iy this ring becomes $\mathbb C[u,v]/(uv-1)=\mathbb C[u,1/u]$, which is factorial and even a PID ( Reason: If $A$ is a PID, so is every ring of fractions $A_S$ [...unless it is a field, you Bourbakistas])

No, quotients of polynomial rings are definitely not "almost UFDs".

Any finitely generated ring over $K$ is such a quotient and this means a lot of non UFDs. Said differently, any algebraic variety in affine space over $K$ has as ring of regular functions one of your quotients and in general (as your own example over $\mathbb R$ states) it will not be a UFD.

By the way, your parenthetical remark about the complex case is a bit ambiguous: $\mathbb C[x,y]/(x^2+y^2-1)$ IS a UFD: by the change of variables $u=x+iy,\ v=x-iy$ this ring becomes $\mathbb C[u,v]/(uv-1)=\mathbb C[u,1/u]$, which is factorial and even a PID. (Reason: If $A$ is a PID, so is every ring of fractions $A_S$ [...unless it is a field, you Bourbakistas].)

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Georges Elencwajg
Georges Elencwajg
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No,quotients of polynomial rings are definitely not "almost UFDs".

Any finitely generated ring over K is such a quotient and this means a lot of non UFDs. Said differently, any algebraic variety in affine space over K has as ring of regular functions one of your quotients and in general (as your own example over $\mathbb R$ states) it will not be a UFD.

By the way, your parenthetical remark about the complex case is a bit ambiguous: $\mathbb C[x,y]/(x^2+y^2-1)$ IS a UFD: by the change of variables u=x+iy,v=x-iy this ring becomes $\mathbb C[u,v]/(uv-1)=\mathbb C[u,1/u]$, which is factorial and even a PID ( Reason: If $A$ is a PID, so is every ring of fractions $A_S$ [...unless it is a field, you Bourbakistas])