R is a ring and consider commutative diagram of R -modules with exact rows.

http://images.planetmath.org:8080/ca...8/l2h/img1.png

then I want to show that Gamma is isomorphism if and only if

the sequence

0----->A_1----->A_2+B_1------>B_2----->0

here + is direct sum and teh first homomorphism sends a to \alpha(a),k_1(a)

and the second homomorphism sends (a_2,b) to k_2(a_2)-\beta(b)

here k_1 is homomorphism from A_1 to B_1 and k_2 is homomorphism from

A_2 to B_2.