The numbering part of the theorem environment is in the "\thexxxx" part where xxxx is the name of the environment. For example, the command
\renewcommand{\thetheorem}{A.\arabic{equation}}
makes
\begin{theorem}...\end{theorem}
having "A.equation_number" as the number after the name of the theorem environment.
Using newtheoremstyle command in the amsthm package (automatically included when amsart class is used), one can make more changes to the format of the theorem. The syntax is
\newtheoremstyle{note}% name
{3pt}% Space above
{3pt}% Space below
{}% Body font
{}% Indent amount1
{\itshape}% Theorem head font
{:}% hPunctuation after theorem head
{.5em}% Space after theorem head2
{}% Theorem head spec (can be left empty, meaning `normal')
1Indent amount: empty = no indent, \parindent = normal paragraph indent
2Space after theorem head: { } = normal interword space; \newline = linebreak
For example, the following will change the name of the theorem to be empty and the numbering goes with the equation like "(A.123)", without dot after it, where 123 is a continuation of the equation numbering.
\newtheoremstyle{dotless}{3pt}{3pt}{\itshape}{}{}{)}{ }{}
\theoremstyle{dotless}
\newtheorem{theorem}[equation]{\rm(\ignorespaces}
\renewcommand{\thetheorem}{A.\arabic{equation}}
Here is a link to amsthm usage file.