Because $in_arg is scalar my $in_arg=@_; assigns the number of elements in @_ (1) to $in_arg.

Instead either:

my ($in_arg)=@_;
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or:

my $in_arg=shift;
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Because $in_arg is scalar my $in_arg=@_;

I would say, if I really wanted to say that,

my $in_arg=scalar @_;
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but then I would just say that @array evaluated in scalar context returns the number of elements in @array and that assignment to a scalar imposes scalar context. Or something like that...

Update: since this was downvoted, why don't you actually try?

$ perl -e 'scalar my $in_arg=@_'
Can't modify scalar in scalar assignment at -e line 1, at EOF
Execution of -e aborted due to compilation errors.
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Update2: I read Re: My first module once again and now I realize I read it uncorrectly. I use a theme in which occasionally inline code is not visually distinctive enough, and GrandFather didn't put a suitable interpunction sign, so this added to my confusion and misreading. Sorry, my apologies...

It is printing 11 because you have the:

my $in_arg=@_;
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my ( $in_arg ) = @_;
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my $in_arg = shift;
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/J\

Aside from the answers coping with your actual problem, you don't have really a problem having to do with your "first module" or a module at all, but incidentally while we're here I'll remind you that you probably shouldn't pollute the main namespace in modules, but put an explicit package in them, exporting only what you really need, or better giving the user choose what he really needs to export.

Also incidentally, even if this was just a minimal example, all lowercase modules are reserved for pragmata.

Just clear up the potentially obvious stuff. Your module should be:

########## test_module.pm ##########
package test_module;
use strict;

sub testsub 
{ 
    my $in_arg= shift;
    my $new_arg=$in_arg+10; 
    return($new_arg); 
} 
1;
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#!/usr/bin/perl -w 

use strict; 
package test_module;
use strict;

sub testsub 
{ 
    my $in_arg= shift;
    my $new_arg=$in_arg+10; 
    return($new_arg); 
} 

my $var=10; 
my $new_var=test_module::testsub($var); 

print "$new_var\n";
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here is my code

#!/usr/bin/perl -w
use strict;

my $var=10;
my $new_var=testsub($var);
print "$new_var\n";

sub testsub
{

    my $in_arg=shift;
    my $new_arg=$in_arg+10;
    return($new_arg);
}
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Edit: g0n - corrected code tags