I am trying to prove the following:
If a Noetherian ring $R$ is reduced, then there is an injection from $R$ into a product of fields.
What I know is that every associated prime of $R$ is minimal and for a minimal prime $p$ the localization $R_p$ is field. But I cannot proceed further. I need some help.
Many Thanks.
Since $R$ is Noetherian, the set of minimal primes is finite, say $p_1,\cdots, p_n$ are the minimal primes of $R$. Since $R$ is reduced the natural map $R\to R/p_{1}\times \cdots \times R/p_{n}$ is injective. Now each $R/p_{i}$ is a domain and hence injects into its field of fractions $Q_i$. Hence we have an injection $R\to Q_1\times \cdots \times Q_n$.