I am new to cryptography, and trying to understand the concepts of LWE (learning with errors) formally. I will state my understanding of the definitions, which might be incorrect.
Definitions According to My Understanding
Let $R$ be a finite unital commutative ring equipped with a probability $\mu$ (whose $\sigma$-algebra is discrete). $R$ is said to satisfy the worst-case search LWE assumption if for all polynomial $n$ and all polynomial-time randomized algorithm $S$, the map
\begin{equation}
m\mapsto \min_{s\in R^m} p(m,s)\text{,}
\end{equation}
where
\begin{equation}
p(m,s) = \Pr\{(A,e)\in R^{m\times n(m)} \times R^{n(m)}: S (-sA+e,A)=s\}\text{,}
\end{equation}
is negligible. In the above equation, $A$ is sampled from the uniform probability, and $e$ is sampled from the product probability $\mu^{n(m)}$.
$R$ is said to satisfy the worst-case decision LWE assumption if for all polynomial $n$ and all polynomial-time randomized algorithm $D$, the map
\begin{equation}
m\mapsto \min_{s\in R^m} |p_1(m,s)-p_2(m)|\text{,}
\end{equation}
where
\begin{equation}
\begin{split}
p_1(m,s) &= \Pr\{(A,e)\in R^{m\times n(m)}\times R^{n(m)}: D (-sA+e,A)=1\},\\
p_2(m) &= \Pr\{(b,A)\in R^{n(m)}\times R^{m\times n(m)}\times: D (b,A)=1\}\text{,}
\end{split}
\end{equation}
is negligible. In the above equation, $A$ and $b$ are sampled from the uniform probabilities, and $e$ is sampled from the product probability $\mu^{n(m)}$.
Question
I proved that if $R$ is a finite field equipped with a probability $\mu$ (whose $\sigma$-algebra is discrete) and if $R$ satisfies the worst-case search LWE assumption, then $R$ also satisfies the worst-case decision LWE assumption. But how does one prove the converse? All literature I have seen so far just say that it is trivial, but I could not prove it. More explicitly, I need a proof of the following statement:
If $R$ is a finite unital commutative ring equipped with a probability $\mu$ (whose $\sigma$-algebra is discrete) and if $R$ satisfies the worst-case decision LWE assumption, then $R$ also satisfies the worst-case search LWE assumption.
My Attempt
Assume that $R$ satisfies the worst-case decision LWE assumption. Let $n$ be a polynomial and let $S$ be a polynomial-time randomized algorithm (whose inputs are elements in $R^{n(m)}\times R^{m\times n(m)}$ and whose outputs are elements in $R^m$). We need to show that the map
\begin{equation}
m\mapsto \min_{s\in R^m} p(m,s)\text{,}
\end{equation}
where $p(m,s)$ is defined above, is negligible. Given an input $(b,A)\in R^{n(m)}\times R^{m\times n(m)}$ where $b=-sA+e$, $S$ will return some guess $g\in R^m$ that might or might not equal $s$. One can compute $b+gA$ and denote it by $e'$. If $g=s$, then $e'=e$. If $g\neq s$, then $e'=e$ might or might not hold. I do not know what to do now.
