diff --git a/Doc/c-api/exceptions.rst b/Doc/c-api/exceptions.rst
index 49d2f18d4573b0..4ed96f01dbbc3e 100644
--- a/Doc/c-api/exceptions.rst
+++ b/Doc/c-api/exceptions.rst
@@ -602,7 +602,7 @@ Signal Handling
 .. c:function:: int PyErr_CheckSignals()
 
    .. index::
-      module: signal
+      pair: module; signal
       single: SIGINT
       single: KeyboardInterrupt (built-in exception)
 
@@ -633,7 +633,7 @@ Signal Handling
 .. c:function:: void PyErr_SetInterrupt()
 
    .. index::
-      module: signal
+      pair: module; signal
       single: SIGINT
       single: KeyboardInterrupt (built-in exception)
 
@@ -648,7 +648,7 @@ Signal Handling
 .. c:function:: int PyErr_SetInterruptEx(int signum)
 
    .. index::
-      module: signal
+      pair: module; signal
       single: KeyboardInterrupt (built-in exception)
 
    Simulate the effect of a signal arriving. The next time
diff --git a/Doc/c-api/init.rst b/Doc/c-api/init.rst
index 38e324fb6409bc..26762969ef8eba 100644
--- a/Doc/c-api/init.rst
+++ b/Doc/c-api/init.rst
@@ -336,9 +336,9 @@ Initializing and finalizing the interpreter
       single: PyEval_InitThreads()
       single: modules (in module sys)
       single: path (in module sys)
-      module: builtins
-      module: __main__
-      module: sys
+      pair: module; builtins
+      pair: module; __main__
+      pair: module; sys
       triple: module; search; path
       single: PySys_SetArgv()
       single: PySys_SetArgvEx()
@@ -1051,7 +1051,7 @@ code, or when embedding the Python interpreter:
 
    .. deprecated:: 3.9
 
-   .. index:: module: _thread
+   .. index:: pair: module; _thread
 
 
 .. c:function:: int PyEval_ThreadsInitialized()
@@ -1494,9 +1494,9 @@ function. You can create and destroy them using the following functions:
 .. c:function:: PyThreadState* Py_NewInterpreter()
 
    .. index::
-      module: builtins
-      module: __main__
-      module: sys
+      pair: module; builtins
+      pair: module; __main__
+      pair: module; sys
       single: stdout (in module sys)
       single: stderr (in module sys)
       single: stdin (in module sys)
diff --git a/Doc/c-api/intro.rst b/Doc/c-api/intro.rst
index acd4e033dfbc4b..1a200b919f0e75 100644
--- a/Doc/c-api/intro.rst
+++ b/Doc/c-api/intro.rst
@@ -705,9 +705,9 @@ interpreter can only be used after the interpreter has been initialized.
 
 .. index::
    single: Py_Initialize()
-   module: builtins
-   module: __main__
-   module: sys
+   pair: module; builtins
+   pair: module; __main__
+   pair: module; sys
    triple: module; search; path
    single: path (in module sys)
 
diff --git a/Doc/library/_thread.rst b/Doc/library/_thread.rst
index 122692a428594f..ba9314e46ab6ea 100644
--- a/Doc/library/_thread.rst
+++ b/Doc/library/_thread.rst
@@ -208,7 +208,7 @@ In addition to these methods, lock objects can also be used via the
 
 **Caveats:**
 
-  .. index:: module: signal
+  .. index:: pair: module; signal
 
 * Threads interact strangely with interrupts: the :exc:`KeyboardInterrupt`
   exception will be received by an arbitrary thread.  (When the :mod:`signal`
diff --git a/Doc/library/binascii.rst b/Doc/library/binascii.rst
index 5a0815faa38eac..21960cb7972e6e 100644
--- a/Doc/library/binascii.rst
+++ b/Doc/library/binascii.rst
@@ -6,8 +6,8 @@
               representations.
 
 .. index::
-   module: uu
-   module: base64
+   pair: module; uu
+   pair: module; base64
 
 --------------
 
diff --git a/Doc/library/cmath.rst b/Doc/library/cmath.rst
index 5ed7a09b3e9db2..b17d58e1cc0ce1 100644
--- a/Doc/library/cmath.rst
+++ b/Doc/library/cmath.rst
@@ -301,7 +301,7 @@ Constants
    .. versionadded:: 3.6
 
 
-.. index:: module: math
+.. index:: pair: module; math
 
 Note that the selection of functions is similar, but not identical, to that in
 module :mod:`math`.  The reason for having two modules is that some users aren't
diff --git a/Doc/library/copy.rst b/Doc/library/copy.rst
index a8bc2fa55ea8c3..8f32477ed508c3 100644
--- a/Doc/library/copy.rst
+++ b/Doc/library/copy.rst
@@ -68,7 +68,7 @@ Shallow copies of dictionaries can be made using :meth:`dict.copy`, and
 of lists by assigning a slice of the entire list, for example,
 ``copied_list = original_list[:]``.
 
-.. index:: module: pickle
+.. index:: pair: module; pickle
 
 Classes can use the same interfaces to control copying that they use to control
 pickling.  See the description of module :mod:`pickle` for information on these
diff --git a/Doc/library/copyreg.rst b/Doc/library/copyreg.rst
index afc3e66f0bf7ac..2a28c043f80723 100644
--- a/Doc/library/copyreg.rst
+++ b/Doc/library/copyreg.rst
@@ -7,8 +7,8 @@
 **Source code:** :source:`Lib/copyreg.py`
 
 .. index::
-   module: pickle
-   module: copy
+   pair: module; pickle
+   pair: module; copy
 
 --------------
 
diff --git a/Doc/library/exceptions.rst b/Doc/library/exceptions.rst
index 18c3f47dddc079..aee1cb5cc6a475 100644
--- a/Doc/library/exceptions.rst
+++ b/Doc/library/exceptions.rst
@@ -318,7 +318,7 @@ The following exceptions are the exceptions that are usually raised.
 .. exception:: OSError([arg])
                OSError(errno, strerror[, filename[, winerror[, filename2]]])
 
-   .. index:: module: errno
+   .. index:: pair: module; errno
 
    This exception is raised when a system function returns a system-related
    error, including I/O failures such as "file not found" or "disk full"
diff --git a/Doc/library/fnmatch.rst b/Doc/library/fnmatch.rst
index 46bf0fc2848058..aed8991d44772f 100644
--- a/Doc/library/fnmatch.rst
+++ b/Doc/library/fnmatch.rst
@@ -8,7 +8,7 @@
 
 .. index:: single: filenames; wildcard expansion
 
-.. index:: module: re
+.. index:: pair: module; re
 
 --------------
 
@@ -38,7 +38,7 @@ special characters used in shell-style wildcards are:
 For a literal match, wrap the meta-characters in brackets.
 For example, ``'[?]'`` matches the character ``'?'``.
 
-.. index:: module: glob
+.. index:: pair: module; glob
 
 Note that the filename separator (``'/'`` on Unix) is *not* special to this
 module.  See module :mod:`glob` for pathname expansion (:mod:`glob` uses
diff --git a/Doc/library/functions.rst b/Doc/library/functions.rst
index 47e3880129595e..ddb918de115fcf 100644
--- a/Doc/library/functions.rst
+++ b/Doc/library/functions.rst
@@ -1340,7 +1340,7 @@ are always available.  They are listed here in alphabetical order.
       single: I/O control; buffering
       single: binary mode
       single: text mode
-      module: sys
+      pair: module; sys
 
    See also the file handling modules, such as :mod:`fileinput`, :mod:`io`
    (where :func:`open` is declared), :mod:`os`, :mod:`os.path`, :mod:`tempfile`,
@@ -1987,7 +1987,7 @@ are always available.  They are listed here in alphabetical order.
 
    .. index::
       statement: import
-      module: builtins
+      pair: module; builtins
 
    .. note::
 
diff --git a/Doc/library/http.client.rst b/Doc/library/http.client.rst
index abdc6b447a8b61..eb8c1e198e2b09 100644
--- a/Doc/library/http.client.rst
+++ b/Doc/library/http.client.rst
@@ -10,7 +10,7 @@
    pair: HTTP; protocol
    single: HTTP; http.client (standard module)
 
-.. index:: module: urllib.request
+.. index:: pair: module; urllib.request
 
 --------------
 
diff --git a/Doc/library/internet.rst b/Doc/library/internet.rst
index ff58dcf4d89c36..681769a4820dba 100644
--- a/Doc/library/internet.rst
+++ b/Doc/library/internet.rst
@@ -9,7 +9,7 @@ Internet Protocols and Support
    single: Internet
    single: World Wide Web
 
-.. index:: module: socket
+.. index:: pair: module; socket
 
 The modules described in this chapter implement internet protocols and  support
 for related technology.  They are all implemented in Python. Most of these
diff --git a/Doc/library/locale.rst b/Doc/library/locale.rst
index f726f8397c9648..f2abb3638a141f 100644
--- a/Doc/library/locale.rst
+++ b/Doc/library/locale.rst
@@ -16,7 +16,7 @@ functionality. The POSIX locale mechanism allows programmers to deal with
 certain cultural issues in an application, without requiring the programmer to
 know all the specifics of each country where the software is executed.
 
-.. index:: module: _locale
+.. index:: pair: module; _locale
 
 The :mod:`locale` module is implemented on top of the :mod:`_locale` module,
 which in turn uses an ANSI C locale implementation if available.
@@ -464,7 +464,7 @@ The :mod:`locale` module defines the following exception and functions:
 
 .. data:: LC_CTYPE
 
-   .. index:: module: string
+   .. index:: pair: module; string
 
    Locale category for the character type functions.  Depending on the settings of
    this category, the functions of module :mod:`string` dealing with case change
diff --git a/Doc/library/marshal.rst b/Doc/library/marshal.rst
index 24f9dc1689da4a..0556f19699dc15 100644
--- a/Doc/library/marshal.rst
+++ b/Doc/library/marshal.rst
@@ -15,8 +15,8 @@ undocumented on purpose; it may change between Python versions (although it
 rarely does). [#]_
 
 .. index::
-   module: pickle
-   module: shelve
+   pair: module; pickle
+   pair: module; shelve
 
 This is not a general "persistence" module.  For general persistence and
 transfer of Python objects through RPC calls, see the modules :mod:`pickle` and
diff --git a/Doc/library/os.path.rst b/Doc/library/os.path.rst
index 96bcb48ad7d126..7881c52db87090 100644
--- a/Doc/library/os.path.rst
+++ b/Doc/library/os.path.rst
@@ -159,7 +159,7 @@ the :mod:`glob` module.)
    On Unix and Windows, return the argument with an initial component of ``~`` or
    ``~user`` replaced by that *user*'s home directory.
 
-   .. index:: module: pwd
+   .. index:: pair: module; pwd
 
    On Unix, an initial ``~`` is replaced by the environment variable :envvar:`HOME`
    if it is set; otherwise the current user's home directory is looked up in the
diff --git a/Doc/library/os.rst b/Doc/library/os.rst
index 76623c6305432a..641e289e77c518 100644
--- a/Doc/library/os.rst
+++ b/Doc/library/os.rst
@@ -1284,7 +1284,7 @@ or `the MSDN <https://msdn.microsoft.com/en-us/library/z0kc8e3z.aspx>`_ on Windo
 
 .. function:: openpty()
 
-   .. index:: module: pty
+   .. index:: pair: module; pty
 
    Open a new pseudo-terminal pair. Return a pair of file descriptors
    ``(master, slave)`` for the pty and the tty, respectively. The new file
@@ -2890,7 +2890,7 @@ features:
    possible and call :func:`lstat` on the result. This does not apply to
    dangling symlinks or junction points, which will raise the usual exceptions.
 
-   .. index:: module: stat
+   .. index:: pair: module; stat
 
    Example::
 
diff --git a/Doc/library/pdb.rst b/Doc/library/pdb.rst
index 8a386aa77368f2..74bffef5562ae1 100644
--- a/Doc/library/pdb.rst
+++ b/Doc/library/pdb.rst
@@ -20,8 +20,8 @@ supports post-mortem debugging and can be called under program control.
 
 .. index::
    single: Pdb (class in pdb)
-   module: bdb
-   module: cmd
+   pair: module; bdb
+   pair: module; cmd
 
 The debugger is extensible -- it is actually defined as the class :class:`Pdb`.
 This is currently undocumented but easily understood by reading the source.  The
diff --git a/Doc/library/posix.rst b/Doc/library/posix.rst
index ec04b0dcfc162f..0413f9d02a8d57 100644
--- a/Doc/library/posix.rst
+++ b/Doc/library/posix.rst
@@ -11,7 +11,7 @@ This module provides access to operating system functionality that is
 standardized by the C Standard and the POSIX standard (a thinly disguised Unix
 interface).
 
-.. index:: module: os
+.. index:: pair: module; os
 
 **Do not import this module directly.**  Instead, import the module :mod:`os`,
 which provides a *portable* version of this interface.  On Unix, the :mod:`os`
diff --git a/Doc/library/pwd.rst b/Doc/library/pwd.rst
index 98f3c45e29cbcb..7cafc66fd7e93c 100644
--- a/Doc/library/pwd.rst
+++ b/Doc/library/pwd.rst
@@ -39,7 +39,7 @@ raised if the entry asked for cannot be found.
 
 .. note::
 
-   .. index:: module: crypt
+   .. index:: pair: module; crypt
 
    In traditional Unix the field ``pw_passwd`` usually contains a password
    encrypted with a DES derived algorithm (see module :mod:`crypt`).  However most
diff --git a/Doc/library/pyexpat.rst b/Doc/library/pyexpat.rst
index d6581e21b01c0e..935e872480efda 100644
--- a/Doc/library/pyexpat.rst
+++ b/Doc/library/pyexpat.rst
@@ -33,7 +33,7 @@ can be set to handler functions.  When an XML document is then fed to the
 parser, the handler functions are called for the character data and markup in
 the XML document.
 
-.. index:: module: pyexpat
+.. index:: pair: module; pyexpat
 
 This module uses the :mod:`pyexpat` module to provide access to the Expat
 parser.  Direct use of the :mod:`pyexpat` module is deprecated.
diff --git a/Doc/library/runpy.rst b/Doc/library/runpy.rst
index 501f4ddf5a3e3f..42ed8c253b8027 100644
--- a/Doc/library/runpy.rst
+++ b/Doc/library/runpy.rst
@@ -30,7 +30,7 @@ The :mod:`runpy` module provides two functions:
 .. function:: run_module(mod_name, init_globals=None, run_name=None, alter_sys=False)
 
    .. index::
-      module: __main__
+      pair: module; __main__
 
    Execute the code of the specified module and return the resulting module
    globals dictionary. The module's code is first located using the standard
@@ -101,7 +101,7 @@ The :mod:`runpy` module provides two functions:
 .. function:: run_path(path_name, init_globals=None, run_name=None)
 
    .. index::
-      module: __main__
+      pair: module; __main__
 
    Execute the code at the named filesystem location and return the resulting
    module globals dictionary. As with a script name supplied to the CPython
diff --git a/Doc/library/shelve.rst b/Doc/library/shelve.rst
index a50fc6f0bf77b2..dc87af398ed757 100644
--- a/Doc/library/shelve.rst
+++ b/Doc/library/shelve.rst
@@ -6,7 +6,7 @@
 
 **Source code:** :source:`Lib/shelve.py`
 
-.. index:: module: pickle
+.. index:: pair: module; pickle
 
 --------------
 
@@ -95,8 +95,8 @@ Restrictions
 ------------
 
   .. index::
-     module: dbm.ndbm
-     module: dbm.gnu
+     pair: module; dbm.ndbm
+     pair: module; dbm.gnu
 
 * The choice of which database package will be used (such as :mod:`dbm.ndbm` or
   :mod:`dbm.gnu`) depends on which interface is available.  Therefore it is not
diff --git a/Doc/library/site.rst b/Doc/library/site.rst
index 4a88013f1d6ed2..a4efefd587a609 100644
--- a/Doc/library/site.rst
+++ b/Doc/library/site.rst
@@ -109,7 +109,7 @@ directory precedes the :file:`foo` directory because :file:`bar.pth` comes
 alphabetically before :file:`foo.pth`; and :file:`spam` is omitted because it is
 not mentioned in either path configuration file.
 
-.. index:: module: sitecustomize
+.. index:: pair: module; sitecustomize
 
 After these path manipulations, an attempt is made to import a module named
 :mod:`sitecustomize`, which can perform arbitrary site-specific customizations.
@@ -121,7 +121,7 @@ with :file:`pythonw.exe` on Windows (which is used by default to start IDLE),
 attempted output from :mod:`sitecustomize` is ignored.  Any other exception
 causes a silent and perhaps mysterious failure of the process.
 
-.. index:: module: usercustomize
+.. index:: pair: module; usercustomize
 
 After this, an attempt is made to import a module named :mod:`usercustomize`,
 which can perform arbitrary user-specific customizations, if
diff --git a/Doc/library/socket.rst b/Doc/library/socket.rst
index c8ca555700a3c9..150c1f93afd967 100644
--- a/Doc/library/socket.rst
+++ b/Doc/library/socket.rst
@@ -1916,7 +1916,7 @@ to sockets.
 .. method:: socket.setsockopt(level, optname, None, optlen: int)
    :noindex:
 
-   .. index:: module: struct
+   .. index:: pair: module; struct
 
    Set the value of the given socket option (see the Unix manual page
    :manpage:`setsockopt(2)`).  The needed symbolic constants are defined in the
diff --git a/Doc/library/stdtypes.rst b/Doc/library/stdtypes.rst
index f6662b4336c276..4c0a213dcbea26 100644
--- a/Doc/library/stdtypes.rst
+++ b/Doc/library/stdtypes.rst
@@ -330,7 +330,7 @@ Notes:
 
 (3)
    .. index::
-      module: math
+      pair: module; math
       single: floor() (in module math)
       single: ceil() (in module math)
       single: trunc() (in module math)
@@ -1600,7 +1600,7 @@ String Methods
 --------------
 
 .. index::
-   module: re
+   pair: module; re
 
 Strings implement all of the :ref:`common <typesseq-common>` sequence
 operations, along with the additional methods described below.
@@ -2511,7 +2511,7 @@ Binary Sequence Types --- :class:`bytes`, :class:`bytearray`, :class:`memoryview
    object: bytes
    object: bytearray
    object: memoryview
-   module: array
+   pair: module; array
 
 The core built-in types for manipulating binary data are :class:`bytes` and
 :class:`bytearray`. They are supported by :class:`memoryview` which uses
@@ -5378,7 +5378,7 @@ Type Objects
 
 .. index::
    builtin: type
-   module: types
+   pair: module; types
 
 Type objects represent the various object types.  An object's type is accessed
 by the built-in function :func:`type`.  There are no special operations on
diff --git a/Doc/reference/compound_stmts.rst b/Doc/reference/compound_stmts.rst
index 0731589e4cadb8..4bdecbf243650f 100644
--- a/Doc/reference/compound_stmts.rst
+++ b/Doc/reference/compound_stmts.rst
@@ -297,7 +297,7 @@ traceback attached to them, they form a reference cycle with the stack frame,
 keeping all locals in that frame alive until the next garbage collection occurs.
 
 .. index::
-   module: sys
+   pair: module; sys
    object: traceback
 
 Before an :keyword:`!except` clause's suite is executed,
diff --git a/Doc/reference/datamodel.rst b/Doc/reference/datamodel.rst
index c35bf4016a28d2..9d3ea3f78de16f 100644
--- a/Doc/reference/datamodel.rst
+++ b/Doc/reference/datamodel.rst
@@ -377,7 +377,7 @@ Sequences
          (and hence unhashable), byte arrays otherwise provide the same interface
          and functionality as immutable :class:`bytes` objects.
 
-      .. index:: module: array
+      .. index:: pair: module; array
 
       The extension module :mod:`array` provides an additional example of a
       mutable sequence type, as does the :mod:`collections` module.
@@ -451,8 +451,8 @@ Mappings
       section :ref:`dict`).
 
       .. index::
-         module: dbm.ndbm
-         module: dbm.gnu
+         pair: module; dbm.ndbm
+         pair: module; dbm.gnu
 
       The extension modules :mod:`dbm.ndbm` and :mod:`dbm.gnu` provide
       additional examples of mapping types, as does the :mod:`collections`
@@ -909,7 +909,7 @@ Class instances
 I/O objects (also known as file objects)
    .. index::
       builtin: open
-      module: io
+      pair: module; io
       single: popen() (in module os)
       single: makefile() (socket method)
       single: sys.stdin
diff --git a/Doc/reference/executionmodel.rst b/Doc/reference/executionmodel.rst
index a264015cbf4049..8917243999d399 100644
--- a/Doc/reference/executionmodel.rst
+++ b/Doc/reference/executionmodel.rst
@@ -151,7 +151,7 @@ to previously bound variables in the nearest enclosing function scope.
 :exc:`SyntaxError` is raised at compile time if the given name does not
 exist in any enclosing function scope.
 
-.. index:: module: __main__
+.. index:: pair: module; __main__
 
 The namespace for a module is automatically created the first time a module is
 imported.  The main module for a script is always called :mod:`__main__`.
diff --git a/Doc/reference/toplevel_components.rst b/Doc/reference/toplevel_components.rst
index 319c9de484241e..ee472ace6e2129 100644
--- a/Doc/reference/toplevel_components.rst
+++ b/Doc/reference/toplevel_components.rst
@@ -21,9 +21,9 @@ Complete Python programs
 .. index:: single: program
 
 .. index::
-   module: sys
-   module: __main__
-   module: builtins
+   pair: module; sys
+   pair: module; __main__
+   pair: module; builtins
 
 While a language specification need not prescribe how the language interpreter
 is invoked, it is useful to have a notion of a complete Python program.  A
@@ -38,7 +38,7 @@ the next section.
 
 .. index::
    single: interactive mode
-   module: __main__
+   pair: module; __main__
 
 The interpreter may also be invoked in interactive mode; in this case, it does
 not read and execute a complete program but reads and executes one statement
diff --git a/Doc/tools/extensions/pyspecific.py b/Doc/tools/extensions/pyspecific.py
index 39c7c42e66f9a0..12d3e382c87318 100644
--- a/Doc/tools/extensions/pyspecific.py
+++ b/Doc/tools/extensions/pyspecific.py
@@ -689,7 +689,7 @@ def patch_pairindextypes(app) -> None:
         # away from this, we need Sphinx to believe that these values don't
         # exist, by deleting them when using the gettext builder.
 
-        # pairindextypes.pop('module', None)
+        pairindextypes.pop('module', None)
         # pairindextypes.pop('keyword', None)
         # pairindextypes.pop('operator', None)
         # pairindextypes.pop('object', None)
@@ -697,10 +697,6 @@ def patch_pairindextypes(app) -> None:
         # pairindextypes.pop('statement', None)
         # pairindextypes.pop('builtin', None)
 
-        # there needs to be at least one statement in this block, will be
-        # removed when the first of the below is uncommented.
-        pass
-
 
 def setup(app):
     app.add_role('issue', issue_role)
diff --git a/Doc/tutorial/inputoutput.rst b/Doc/tutorial/inputoutput.rst
index 3581b3727a53ea..aeacc1559a228c 100644
--- a/Doc/tutorial/inputoutput.rst
+++ b/Doc/tutorial/inputoutput.rst
@@ -466,7 +466,7 @@ Reference for a complete guide to file objects.
 Saving structured data with :mod:`json`
 ---------------------------------------
 
-.. index:: module: json
+.. index:: pair: module; json
 
 Strings can easily be written to and read from a file.  Numbers take a bit more
 effort, since the :meth:`read` method only returns strings, which will have to
diff --git a/Doc/tutorial/modules.rst b/Doc/tutorial/modules.rst
index 4daafa49a34d2e..3bd034bcc9703f 100644
--- a/Doc/tutorial/modules.rst
+++ b/Doc/tutorial/modules.rst
@@ -264,7 +264,7 @@ Some tips for experts:
 Standard Modules
 ================
 
-.. index:: module: sys
+.. index:: pair: module; sys
 
 Python comes with a library of standard modules, described in a separate
 document, the Python Library Reference ("Library Reference" hereafter).  Some
@@ -345,7 +345,7 @@ Without arguments, :func:`dir` lists the names you have defined currently::
 
 Note that it lists all types of names: variables, modules, functions, etc.
 
-.. index:: module: builtins
+.. index:: pair: module; builtins
 
 :func:`dir` does not list the names of built-in functions and variables.  If you
 want a list of those, they are defined in the standard module