Editing Projective module (section)

== Elementary examples and properties ==
The following properties of projective modules are quickly deduced from any of the above (equivalent) definitions of projective modules:
* Direct sums and direct summands of projective modules are projective.
* If {{math|1=''e'' = ''e''<sup>2</sup>}} is an [[idempotent (ring theory)|idempotent]] in the ring {{math|''R''}}, then {{math|''Re''}} is a projective left module over ''R''.

Let <math>R = R_1 \times R_2</math> be the [[direct product]] of two rings <math>R_1</math> and <math>R_2,</math> which is a ring with operations defined componentwise. Let <math>e_1=(1,0)</math> and <math>e_2=(0,1).</math> Then <math>e_1</math> and <math>e_2</math> are idempotents, and belong to the [[centre of a ring|centre]] of <math>R.</math> The [[two-sided ideal]]s <math>Re_1</math> and <math>Re_2</math> are projective modules, since their direct sum (as {{mvar|R}}-modules) equals the free {{mvar|R}}-module {{mvar|R}}. However, if <math>R_1</math> and <math>R_2</math> are nontrivial, then they are not free as modules over <math>R</math>. For instance <math>\mathbb{Z}/2\mathbb{Z}</math> is projective but not free over <math>\mathbb{Z}/6\mathbb{Z}</math>.