Let $R$ be an integral domain and let $K$ be its field of fractions. Consider the ring $T=\mathbf Q\otimes_\mathbf Z R$. We have a homomorphism $$\varphi\colon T\longrightarrow K$$ $$a\otimes r\longmapsto ar\qquad a\in \mathbf Q,r\in R.$$ Under what conditions on $R$ is $\varphi$ (a) injective, (b) surjective? When is $T$ a field? When do $T$ and $K$ coincide?
2026-04-08 16:30:24.1775665824
When does the extension of scalars to $\mathbf Q$ give the field of fractions?
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