A Polynomial Approach to Linear Algebra by Paul A. Fuhrmann

M 1 + .. + Mk defined by is a module isomorphism. 5. Let M be a module and M, submodules. Show that if M = M 1 + . . +Mk and M 1 nM2 (M 1 + M 2 ) n M3 = 0 0 then M = M 1 ill . ill M k . 6. Let M be a module and K, L submodules. (a) Show that (K +L)IK ~ LI(KnL). 32 1. Preliminaries (b) If K c L eM, then MIL:::= (MIK)/(LIK). 7. A module M over a ring R is called free if it is the zero-module or has a basis. Show that if M is a free module over a principal ideal domain R having n basis elements and N a submodule, then N is free and has at most n basis elements.

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