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All rings are commutative and unital

Q1: what means notation $$A\cong A_1\times\ldots\times A_n?$$ Is it true that elements of $A_1\times\ldots\times A_n$ are collection of elements of $A_1,\ldots ,A_n$ with termwise multiplication and addition? What is the difference between $$A_1\times\ldots\times A_n$$ and $$A_1\oplus\ldots\oplus A_n?$$

Q2: Let's $\{\mathfrak{m}_i\}_{i\in I}$ is the set of all maximal ideals of ring $A$. Is it true that $\cup_{i\in I}\mathfrak{m}_i$ consists of all non-invertible elements and $A-\{\mathfrak{m}_i\}_{i\in I}$ consists of all invertible elements?

Thanks a lot!

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    For 1: There is essentially no difference between the direct product and direct sum of rings for a finite index set.2012-09-16
  • 2
    For 2: Yes, and I think you must apply Zorn's lemma.2012-09-16

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