Special linear group of a quotient

Let J be a non-trivial ideal of a commutative ring A. The canonical map from A to the quotient A/J induces a homomorphism $\varphi : SL_n(A) \to SL_n(A/J)$. In general $\varphi$ is not surjective (for an example you can look here, comment 7 octobre 2010).

Question: Is there an example of non-surjectiveness using only basics known to an undergrad?

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