Description: With real numbers $\mathbb R$, $\mathbb R[x]/(x^2+1)$ OR the set of real matrices of form $\left[\begin{smallmatrix}a&b\\-b&a\end{smallmatrix}\right]$ with determinant greater than $0$. OR the real Clifford algebra for signature (-)

Notes:

Keywords quotient ring subring matrix ring

Reference(s):

This ring has the following properties:

ACC principal
algebraically closed field
Artinian
atomic domain
Bezout domain
Bezout ring
characteristic 0 field
clean
Cohen-Macaulay
coherent
connected
continuous
Dedekind domain
distributive
domain
dual
Euclidean domain
field
finitely pseudo Frobenius
Frobenius
GCD domain
Gorenstein
Jacobson (Hilbert)
Kasch
Krull domain
local
local complete intersection ring
Mori domain
Noetherian
normal
normal domain
perfect
perfect field
principal ideal domain
principal ideal ring
Prufer domain
pseudo Frobenius
Pythagorean field
quadratically closed field
rad-nil
reduced
regular
regular local
Schreier domain
self-injective
semilocal
semiperfect
semiprimary
semiprimitive
semiregular
semisimple
serial
stable range 1
strongly pi regular
unique factorization domain
valuation
von Neumann regular

The ring lacks the following properties:

We don't know if the ring has or lacks the following properties: