**Problem.** 1) Let be a commutative ring with unity and some ideals of If there exists a surjective -module homomorphism then

2) Show that the result in 1) may not be true in noncommutative rings.

**Solution**. 1) We have for some Now if then and thus

So

2) Let be the ring of matrices with, say, real entries. Let and See that are left ideals of and that is not contained in Now define in this way: for any we define It is easy to see that is a well-defined -module homomorphism. Also, is surjective because if and then because

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