"I think there a lot of syntax errors ...

Welcome. Your running with strict, warnings and diagnostics, perl will tell you about problems.

my $dir = "usr/bin/Test_Protokoll"
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Should be:

my $dir = "usr/bin/Test_Protokoll";
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If this is the same bet you're talking about, you need to specify the output path/file, bet input.file outdir/out.file -m. You'll also have seen:

"Numeric variables with more than one digit may not start with '0' at bet.pl line 9 (#1)"

So don't do that. You improperly create @filter and then never use it. Consider:

#!/usr/bin/perl

use strict;
use warnings;
use diagnostics;

my $dir = 'usr/bin/Test_Protokoll';
my @filters = ('001', '002', '003');
foreach my $filter ( @filters ){
    print "filter: $filter\n";    
    system("bet SYNT_MPRAGE_T1_PD_B0corr YNT_MPRAGE_T1_PD_B0corr_brain
+ -m");
}
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You don't say what you want to do with these filters, and perhaps you want to include $dir as the output path to your bet call. See also $?