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 uoncommututive rings need not have a unique rank) Let M be the Zmodule Z X Z X ..., and let R = EndZUi/I). Dene $1, 462 E R by 1(a1, (12, (153, ...) = (a1j a3, @5, ..) and (352(61, (L2, (131 ..) = (G2, 034, (155, ....) (a) Prove that {(351, $2} is a free basis of the left Rmodule R. [Dene the maps 1&1 and #12 by 1(u1,u2, ...) = (a1, 0,u2,0, ...) and 2(u1,u2, ...) = (0,u1,0, a2, ....) Verify that au = 1, etude = 0 = qty/J1, and nch + #22962 = 1- Use these relations to show that $51, (252 are independent and generate R as a left Rmodule.] (b) Use part (a) to prove that R E\" R2, and deduce that R E R" for any n E N