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I am having trouble coming up with an example to an old exam I am using to help study for some linear algebra qualifiers.

Let $R$ be commutative ring and set $M = R \times R$. Considering $M^2$ as a $R$ module is there an example of a surjective $R$-module homomorphism from $f: M^2 \rightarrow M$ so that the matrix corresponding to the linear transformation $f$ contains only entries which are divisors of zero.

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    Surely the OP means to keep surjectivity. Otherwise the question is kinda pointless. But there are several other typos, too. $M^2$ is an $R$-module. And I'm not sure whether the question is about a mapping from $R^4\to R^2$ or $R^2\to R$. Luckily my hint works all the same. BTW, I would use the term *linear transformation* only between vector spaces, and use *$R$-module homomorphism* here.2011-11-06

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Hint: Let $R=\mathbf{Z}/6\mathbf{Z}$.

Edit: Hint2: Consider mapping $R^2\to R$ that sends $(m,n)\mapsto(2m+3n)$.

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    @user7980: What Gerry says. Now spelled out...2011-11-07