Let ∅ be the element of End((Z x Z, +)) given in Example 24.2. That example showed that ∅ is a right divisor of 0. Show that ∅ is also a left divisor of 0.

Data from Example 24.2

Consider the abelian group (Z x Z, +) discussed in Section 11. It is straightforward to verify that two elements of End( (Z x Z, +)) are ∅ and ψ defined by ∅((m, n)) = (m + n, 0) and ψ((m, n)) = (0, n).

Note that ∅ maps everything onto the first factor of Z x Z, and ψ collapses the first factor. Thus (ψ∅(m, n) = ψ(m + n, 0) = (0, 0).  while  (∅ψ)(m, n) = ∅(0, n) = (n, 0).  Hence ∅ ψ ≠ ψ∅.