Idempotents

We are going to decompose the ring of Witt vectors $\mathcal W_1(A)$. Before doing that, we review facts on idempotents. Recall that an element $x$ of a ring is said to be idempotent if $x^2 =x$.

THEOREM 9.1   Let $R$ be a commutative ring. Let $e\in R$ be an idempotent. Then:

1. $\tilde e=1-e$ is also an idempotent. (We call it the complementary idempotent of $e$.)
2. $e,\tilde e$ satisfies the following relations:
   $$e^2=1,\quad {\tilde e} ^2=1, \quad e \tilde e = 0.$$

3. $R$ admits an direct product decomposition:
   $$R= (R e ) \times ( R \tilde e )$$

DEFINITION 9.2   For any ring $R$, we define a partial order on the idempotents of if as follows:

$$e \succeq f \ \iff \ e f=f$$

It is easy to verify that the relation $\succeq$ is indeed a partial order. We note also that, having defined the order on the idempotents, for any given family $\{e_\lambda \}$ of idempotents we may refer to its “supremum” $\vee e_\lambda$ and its“infimum” $\wedge e_\lambda$. (We are not saying that they always exist: they may or may not exist. ) When the ring $R$ is topologized, then we may also discuss them by using limits,