Let R be a ring with identity, not necessarily commutative. An element e R is said to be idempotent if e^2 = e.

1) Let x R. Show that the set RxR = {rxs : r, s R} is an ideal in R.

2) Let R be commutative. Show that e R is idempotent if and only if ReR R(1 e)R = 0 the 0 ideal. (Hint: Consider the product e(1 e) and use the idea from part (d)).

3) Let R be a ring, not necessarily commutative. Consider the subset eRe = {ere : r R}. Check that eRe is a ring, with addition and multiplication as defined in R, with additive identity 0, but with multiplicative identity e. Why is this not considered a subring of R?