Ring of trigonometric polynomials

Definition
Let $$R$$ be a commutative unital ring. The ring of trigonometric polynomials or ring of circular polynomials over $$R$$ is defined as the ring $$R[x,y]/(x^2 + y^2 - 1)$$. Equivalently, it is the ring $$R[\cos t, \sin t]$$ where $$\cos t$$ and $$\sin t$$ are subject to the usual relation $$\cos^2 t + \sin^2 t = 1$$.

Affine ring
The ring of trigonometric polynomials is an affine ring over $$R$$. In particular, when $$R$$ is a field, the ring of trigonometric polynomials is an affine ring over a field.

Thus, the ring $$R[x,y]/(x^2 + y^2 - 1)$$ is a Noetherian ring whenever $$R$$ is a Noetherian ring.

Integral domain
When $$R$$ has characteristic two, we have $$x^2 + y^2 - 1 = (x + y + 1)^2$$, so the quotient is the ring $$R[x,y]/(x+y+1)^2$$ which is a local ring with unique maximal ideal generated by $$x + y + 1$$.

On the other hand, when $$R$$ is a field of characteristic not equal to two, the polynomial $$x^2 + y^2 -1$$ is irreducible over $$R[x,y]$$. Thus, the quotient $$R[x,y]/(x^2 + y^2 - 1)$$.

Unique factorization domain
For $$R$$ a field of characteristic not equal to two, the ring of trigonometric polynomials over $$R$$ is a unique factorization domain if and only if $$-1$$ is a square in the field.