Answer Happy

Question is in Picture. This is from Abstract Algebra: Modules.
Please answer if you are familiar with the topic and can answer in
a clear and detailed way. The textbook used is: Abstract Algebra -
3rd Edition by David S. Dummit, Richard M. Foote. Thank you!
10. (Free modules over noncommutative rings need not have a unique rank) Let M be the Z-module Z Z X ..., and let R= Endz(M). Define 01, 02 € R by 01(a1, A2, A3, ...) = (a1, Q3, Q5, ...) and 02(21, 22, 23, ...) = (a2, 24, 26, ...). = = a - (a) Prove that {01, 02} is a free basis of the left R-module R. (Define the maps 41 and 42 by V1(a1, Q2, ...) = (a1,0, 22, 0, ...) and 42 (a1, A2, ...) (0, 01, 0, A2, ...). Verify that Oiti = 1, 0142 = 0) = 0201, and 4101 +4202 1. Use these relations to show that 01, 02 are independent and generate R as a left R-module.] (b) Use part (a) to prove that R = R², and deduce that R = RM for any n E N.