Let $A$ be a commutative ring with identity. We say that $A$ is a real ring if every sum of squares of non-zero elements of $A$ is not zero. It is well-known that if $A$ is a real ring and an integral domain, then the quotient field of $A$ is an orderable field. My question: Give an example of a real ring which is not integral domain.
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Just take a direct product of real rings, for example $\Bbb Z\times\Bbb Z$.
Angina Seng
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$\mathbb{Z}$ x $\mathbb{Z}$ is good example for your question.$(1,0).(0,1)=(0,0)$ – 1ENİGMA1 Nov 28 '17 at 07:29