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Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the quotient $A/I$?

I asked this on MSE and received some interesting comments, but I would still like to know the big picture and perhaps some examples. Taking e.g $\langle x\rangle\vartriangleleft A[x]$, the localization $A[x]_{1+\langle x\rangle}$ seems like it takes $x$ to something "very infinitesimal". I don't recognize this ring as the functions on some familiar "space" as e.g the dual numbers are the functions on an infinitesimal line segment. What is the idea here?

Added. I thought about viewing the localization as a subring of the ring of formal power series comprised of the polynomials along with "some" power series, namely those of the form $\sum_k x^k f^k=(1-xf)^{-1}$. Don't really know where to go from here though. I also received the observation that $A[x]_{1+\langle x\rangle}$ inverts the polynomials whose value at zero is invertible, which remains true in the multivariate case. Still I wonder how to interpret the case of a general ideal.

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the quotient $A/I$?

I asked this on MSE and received some interesting comments, but I would still like to know the big picture and perhaps some examples. Taking e.g $\langle x\rangle\vartriangleleft A[x]$, the localization $A[x]_{1+\langle x\rangle}$ seems like it takes $x$ to something "very infinitesimal". I don't recognize this ring as the functions on some familiar "space" as e.g the dual numbers are the functions on an infinitesimal line segment. What is the idea here?

Added. I thought about viewing the localization as a subring of the ring of formal power series comprised of the polynomials along with "some" power series, namely those of the form $\sum_k x^k f^k=(1-xf)^{-1}$. Don't really know where to go from here though. I also received the observation that $A[x]_{1+\langle x\rangle}$ inverts the polynomials whose value at zero is invertible, which remains true in the multivariate case. Still I wonder how to interpret the case of a general ideal.

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the quotient $A/I$?

I asked this on MSE and received some interesting comments, but I would still like to know the big picture and perhaps some examples. Taking e.g $\langle x\rangle\vartriangleleft A[x]$, the localization $A[x]_{1+\langle x\rangle}$ seems like it takes $x$ to something "very infinitesimal". I don't recognize this ring as the functions on some familiar "space" as e.g the dual numbers are the functions on an infinitesimal line segment. What is the idea here?

Added. I thought about viewing the localization as a subring of the ring of formal power series comprised of the polynomials along with "some" power series, namely those of the form $\sum_k x^k f^k=(1-xf)^{-1}$. Don't really know where to go from here though.

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the quotient $A/I$?

I asked this on MSE and received some interesting comments, but I would still like to know the big picture and perhaps some examples. Taking e.g $\langle x\rangle\vartriangleleft A[x]$, the localization $A[x]_{1+\langle x\rangle}$ seems like it takes $x$ to something "very infinitesimal". I don't recognize this ring as the functions on some familiar "space" as e.g the dual numbers are the functions on an infinitesimal line segment. What is the idea here?

Added. I thought about viewing the localization as a subring of the ring of formal power series comprised of the polynomials along with "some" power series, namely those of the form $\sum_k x^k f^k=(1-xf)^{-1}$. Don't really know where to go from here though. I also received the observation that $A[x]_{1+\langle x\rangle}$ inverts the polynomials whose value at zero is invertible, which remains true in the multivariate case. Still I wonder how to interpret the case of a general ideal.

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the quotient $A/I$?

I asked this on MSE and received some interesting comments, but I would still like to know the big picture and perhaps some examples. Taking e.g $\langle x\rangle\vartriangleleft A[x]$, the localization $A[x]_{1+\langle x\rangle}$ seems like it takes $x$ to something "very infinitesimal". I don't recognize this ring as the functions on some familiar "space" as e.g the dual numbers are the functions on an infinitesimal line segment. What is the idea here?

Let $I\vartriangleleft A$ be an ideal of a commutative ring. Consider the submonoid $1+I\subset A$. What is the geometric interpretation of localization at this submonoid? How does it relate to the quotient $A/I$?

I asked this on MSE and received some interesting comments, but I would still like to know the big picture and perhaps some examples. Taking e.g $\langle x\rangle\vartriangleleft A[x]$, the localization $A[x]_{1+\langle x\rangle}$ seems like it takes $x$ to something "very infinitesimal". I don't recognize this ring as the functions on some familiar "space" as e.g the dual numbers are the functions on an infinitesimal line segment. What is the idea here?

Added. I thought about viewing the localization as a subring of the ring of formal power series comprised of the polynomials along with "some" power series, namely those of the form $\sum_k x^k f^k=(1-xf)^{-1}$. Don't really know where to go from here though.