Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ which fails already to be surjective with $n=1$ and $R=\mathbf{Z}$).

In general, there is no reason to expect the projection $\pi$ to be surjective. However, when $R$ is a PID (principal ideal domain), it is not too difficult to show (using elementary divisors) that $\pi$ is surjective.

Q1: Is there some abstract general criterion that will guarantee the surjectivity of $\pi$ ?

Q2: Is $\pi$ surjective if $R$ is a Dedekind domain (Localization a the finite set of primes appearing in the support of $I$ is not good enough, since one also wants the determinant to be equal to $1$) ?

Q3: Is $\pi$ surjective if $R$ is an order of a Dedekind ring $\mathcal{O}$ ?

added the first question is a duplicate of:

When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ which fails already to be surjective with $n=1$ and $R=\mathbf{Z}$).

In general, there is no reason to expect the projection $\pi$ to be surjective. However, when $R$ is a PID (principal ideal domain), it is not too difficult to show (using elementary divisors) that $\pi$ is surjective.

Q1: Is there some abstract general criterion that will guarantee the surjectivity of $\pi$ ?

Q2: Is $\pi$ surjective if $R$ is a Dedekind domain (Localization a the finite set of primes appearing in the support of $I$ is not good enough, since one also wants the determinant to be equal to $1$) ?

Q3: Is $\pi$ surjective if $R$ is an order of a Dedekind ring $\mathcal{O}$ ?

added the first question is a duplicate of:

When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ which fails already to be surjective with $n=1$ and $R=\mathbf{Z}$).

In general, there is no reason to expect the projection $\pi$ to be surjective. However, when $R$ is a PID (principal ideal domain) (with $n\geq 2$), it is not too difficult to show (using elementary divisors) that $\pi$ is surjective.

Q1: Is there some abstract general criterion that will guarantee the surjectivity of $\pi$ ?

Q2: Is $\pi$ surjective if $R$ is a Dedekind domain (Localization a the finite set of primes appearing in the support of $I$ is not good enough, since one also wants the determinant to be a unit in $R$) ?

Q3: Is $\pi$ surjective if $R$ is an order of a Dedekind ring $\mathcal{O}$ ?

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ which fails already to be surjective with $n=1$ and $R=\mathbf{Z}$).

In general, there is no reason to expect the projection $\pi$ to be surjective. However, when $R$ is a PID (principal ideal domain), it is not too difficult to show (using elementary divisors) that $\pi$ is surjective.

Q1: Is there some abstract general criterion that will guarantee the surjectivity of $\pi$ ?

Q2: Is $\pi$ surjective if $R$ is a Dedekind domain (Localization a the finite set of primes appearing in the support of $I$ is not good enough, since one also wants the determinant to be equal to $1$) ?

Q3: Is $\pi$ surjective if $R$ is an order of a Dedekind ring $\mathcal{O}$ ?

added the first question is a duplicate of:

When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:GL_n(R)\rightarrow GL_n(R/I). $$ In general, there is no reason to expect the projection $\pi$ to be surjective. However, when $R$ is a PID (principal ideal domain), it is not too difficult to show (using elementary divisors) that $\pi$ is surjective.

Q1: Is there some abstract general criterion that will guarantee the surjectivity of $\pi$ ?

Q2: Is $\pi$ surjective if $R$ is a Dedekind domain (Localization a the finite set of primes appearing in the support of $I$ is not good enough, since one also wants the determinant to be a unit in $R$) ?

Q3: Is $\pi$ surjective if $R$ is an order of a Dedekind ring $\mathcal{O}$ ?

Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map $$ \pi:SL_n(R)\rightarrow SL_n(R/I) $$ (In the original question I had put $GL_n$ instead of $SL_n$ which fails already to be surjective with $n=1$ and $R=\mathbf{Z}$).

In general, there is no reason to expect the projection $\pi$ to be surjective. However, when $R$ is a PID (principal ideal domain) (with $n\geq 2$), it is not too difficult to show (using elementary divisors) that $\pi$ is surjective.

Q1: Is there some abstract general criterion that will guarantee the surjectivity of $\pi$ ?

Q2: Is $\pi$ surjective if $R$ is a Dedekind domain (Localization a the finite set of primes appearing in the support of $I$ is not good enough, since one also wants the determinant to be a unit in $R$) ?

Q3: Is $\pi$ surjective if $R$ is an order of a Dedekind ring $\mathcal{O}$ ?