for (my $element = 0; $element<$total_element; $element++)
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Don't. Use the following instead:

for my $command (@run) {
...
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This is much saner, as you don't need to know about $total_element and cannot make errors by using $element in a wrong way.

Also, please check the return value from system instead of only querying $?. Use the following:

if (system( $run_cmd ) != 0) {
    print "Running command '$run_cmd' failed: $!/$?\n";
    $err = 1;
};
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Get a better keyboard that doesn't repeat punctuation characters.

Don't use C for loops. Avoid rewriting the same code. Use English prose white space conventions where applicable. A little restructuring gives:

use strict;
use warnings;

my @run;

for my $element (@run) {
    my ($run_cmd, $chdir_cmd) = split ';', $element, 2;
    ch_dir ($chdir_cmd, $patch, '0');

    system ($run_cmd);
    if ($run_cmd =~ /check.pl/) {
        next if $? == 2;
    } elsif ($? == 0) {
        next;
    }

    print "Command Failed: While Executing $run_cmd$!\n";
    $err = 1;
}
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which is of course incomplete and untested.

Argument "cd /and/people/dir" isn't numeric in array element at release_patch.pl line 130. i am getting following warning how to get rid of it???

use eq instead.

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True laziness is hard work

It seems your commands are broken, and you're spending effort trying to fix them at run-time. foo ; cd bar should be cd bar ; foo. You should fix your data before running your program, or at least separately from trying to execute them.

Why is a check.pl dying from SIGINT with no coredump considered successful? Why is a check.pl dying from SIGINT with coredump considered unsuccessful? While is running a program successfully considered an error? You're probably killing the program when you should be closing its input stream with Ctrl-D (default for unix) or Ctrl-Z (Windows).

You're printing $! when its meanlingless. You're not printing $? when its meanlingful.

You are using a C-style loop when a Perl-style counting loop would be simpler. In fact, you are using a C-style loop when a foreach loop would be ideal.

@run is both misnamed (it doesn't contain a run) and incorrectly pluralised (it s singular yet it contains multiple of what it contains).

Basically, that whole snippet should be:

for my $cmd (@cmds) {
   system($cmd)
   die("<$cmd>: Can't spawn: $!\n")                   if $? == -1;
   die("<$cmd> died from signal ", ($? & 0x7F), "\n") if $? & 0x7F;
   die("<$cmd> exited with code ", ($? >> 8), "\n")   if $? >> 8;
}
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Everything extra is you trying to fix bugs made elsewhere, bugs better fixed elsewhere since fixing those bugs have nothing to do with executing commands.

Please read How do I post a question effectively? regarding the title of your post.

Martin

    if ($run_cmd eq 'check.pl')
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Given the usage in the OP's code $total_element is the number of element rather than the largest index - this is why C style for loops are so prone to off by 1 errors. ;-)

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True laziness is hard work