2
$\begingroup$

Let $R$ be a commutative ring with identity. Suppose that $M_1, M_2, ..., M_{2n}$ are distinct maximal ideals in $R$. Let $\bar R=R/M_1M_2\cdots M_{2n} \cong R/M_1 \times R/M_2 \times \cdots\times R/M_{2n}$.

What is the meaning of this statement:

Let $\{ f_1,f_2,...,f_{2n}\}$ be idempotents of $\bar R$ corresponding to the standard basis for $R/M_1 \times R/M_2 \times \cdots\times R/M_{2n}$.

Also, does $f_{n+i}+1=f_i$.

  • 1
    It means the elements which are $1$ in one entry and $0$ in the rest. – 2017-01-13
  • 0
    aha. I got confused about it since there are idempotents other than standard basis. Thanks dude – 2017-01-13

1 Answers 1

1

The term "standard basis" is rather misleading here because you don't actually literally have a basis of anything. They just mean that $f_i$ is the element of the product which is $1$ on the $i$th coordinate and $0$ on every other coordinate. So in particular, it is not true that $f_{n+i}=f_i$ (unless $n=1$ and $R/M_2$ has characteristic $2$).