Unformatted text preview: x 2 + 1) 2 . Determine the elementary divisors of M and the invariant factors of M . 10 If N is a direct summand of M (i.e. M = N ⊕ K , show N is pure in M . 10 If N is a pure submodule of M and ann( x + N ) = ( d ), prove that w can be chosen in x + N such that w + N = x + N and ann( w ) = ( d ). 10 If N is a pure submodule of a ﬁnitely generated module M over a PID D , prove that N is a direct summand of M . You may assume that a ﬁnitely generated module of a PID is a direct product of cyclic modules. 1...
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 Spring '09
 Ring, Ann, Direct sum of modules, Finitely generated module, pure submodule

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