documentclass{amsart}
usepackage{amsthm}
usepackage{zref}% http://ctan.org/pkg/zref

makeatletter
letoldlabellabel
renewcommand{label}[1]{%
zref@labelbylist{#1}{special}% Special label
oldlabel{#1}% Old label
}
newcounter{splabel}
zref@newlist{special}% Create a new property list called special
zref@newprop{section}{arabic{section}}% Section property holds arabic{section}
zref@addprop{special}{section}% Add a section property to special
newcommand*{thmref}[1]{%
  stepcounter{splabel}% Increment local "special label" counter
zref@labelbylist{#1-thesplabel}{special}% Create label
edeftargetsec{zref@extractdefault{#1}{section}{-1}}% Extract target section
edefsourcesec{zref@extractdefault{#1-thesplabel}{section}{-1}}% Extract source section
ifnumtargetsec=sourcesecelsetargetsec.fi%
ref{#1}%
}

newtheorem{dummy}{***}[subsection]% Used so that theorems, definitions, etc can have same counter within subsections
newtheorem{theorem}[dummy]{Theorem}
newtheorem{lemma}[dummy]{Lemma}

theoremstyle{definition}
newtheorem{definition}[dummy]{Definition}
newtheorem{remark}[dummy]{Remark}

renewcommand{thesubsection}{arabic{subsection}}% Custom numbering on subsection to remove section number

begin{document}

section{First Section}
subsection{First Subsection in Section 1}
begin{definition}
    Good definition.
end{definition}

subsection{Second Subsection in Section 1}
begin{remark}
    Interesting remark.
end{remark}
begin{lemma} label{lem:name}
    Useful lemma.
end{lemma}

subsection{Third subsection in Section 1}

begin{theorem} label{thm:name}
    Important theorem.
end{theorem}

section{Section 2}
subsection{Another subsection} There's not much to say here.
subsection{Last Subsection}
 As you can see, Theorem thmref{thm:name}  is very important. Indeed, Lemma~thmref{lem:name} was quite useful.
end{document}