Exercise 5, § 2

Let A be a ring.

a) An element $e$ of A is idempotent if and only if the mapping $x\mapsto xe$ is a projector of the left A-module $A_s$ (resp. the mapping $x\mapsto ex$ is a projector of the right A-module $A_d$).

b) Let $e_1, e_2$ be idempotents in the ring A. Prove that the group Hom$_A(Ae_1,Ae_2)$ is isomorphic to $e_1A\cap Ae_2=e_1Ae_2$ and that the ring End$_A(Ae_1)$ is isomorphic to the ring $e_1Ae_1$.

c) Let $e_1, e_2$ be idempotents in A. Prove that the following properties are equivalent:

(i) $Ae_1= Ae_2$.

(ii) $e_1e_2=e_1$ and $e_2e_1=e_2$.

(iii) $(1-e_1)A = (1-e_2)A$.

When they are satisfied, there exists an invertible element $a\in A$ such that $e_2=$ $ae_1a^{-1}$. d) Let $e_1, e_2$ be two idempotents in A. Prove that the following properties are equivalent:

(i) The left A-modules $Ae_1$ and $Ae_2$ are isomorphic.

(ii) The right A-modules $e_1A$ and $e_2A$ are isomorphic.

(iii) There exist elements $x, y$ of A such that $xy=e_1$ and $yx=e_2$.

When they are satisfied, we say that $e_1$ and $e_2$ are equivalent idempotents.

e) If idempotents in A are equivalent and belong to the center of A, then they are equal.

f ) Let $(e_i)_{i\in I}$ and $(f_i)_{i\in I}$ be finite families of idempotents in A such that $e_ie_j= 0$ and $f_if_j= 0$ for $i\in I,j\in I,i\not=j$ and that $e_i$ is equivalent to $f_i$ for every $i\in I$. Then $e=\sum_{i\in I}e_i$ and $f=\sum_{i\in I}f_i$ are equivalent idempotents in A.