diff --git a/Lib/math/__init__.py b/Lib/math/__init__.py
@@ -0,0 +1,20 @@
+"""
+This module provides access to the mathematical functions
+defined by the C standard.
+"""
+
+from _math import *
+
+# gh-140824: Fix module name for pickle
+def patch_module(objs, module):
+    for obj in objs:
+        if not hasattr(obj, "__module__"):
+            continue
+        obj.__module__ = module
+patch_module([obj for name, obj in globals().items()
+              if not name.startswith('_')], 'math')
+
+from _math_integer import comb, factorial, gcd, isqrt, lcm, perm
+patch_module([comb, factorial, gcd, isqrt, lcm, perm], 'math.integer')
+
+del patch_module
diff --git a/Lib/math/integer.py b/Lib/math/integer.py
@@ -0,0 +1,5 @@
+"""
+This module provides access to integer related mathematical functions.
+"""
+
+from _math_integer import *
diff --git a/Lib/test/test_math.py b/Lib/test/test_math.py
@@ -2788,6 +2788,27 @@ def assertIsNegativeZero(self, value):
         )
 
 
+class MiscTests(unittest.TestCase):
+
+    def test_module_name(self):
+        # gh-140824: _math and _math_integer extensions are exported as math
+        # and math.integer names.
+        math_integer_names = {'comb', 'factorial', 'gcd', 'isqrt', 'lcm', 'perm'}
+        for name in dir(math):
+            if name.startswith('_'):
+                continue
+            obj = getattr(math, name)
+            if not hasattr(obj, '__module__'):
+                continue
+
+            if name in math_integer_names:
+                module = 'math.integer'
+            else:
+                module = 'math'
+            with self.subTest(name=name):
+                self.assertEqual(obj.__module__, module)
+
+
 def load_tests(loader, tests, pattern):
     from doctest import DocFileSuite
     tests.addTest(DocFileSuite(os.path.join("mathdata", "ieee754.txt")))

@@ -2568,6 +2568,7 @@ LIBSUBDIRS=	asyncio \
 		importlib importlib/resources importlib/metadata \
 		json \
 		logging \
+		math \
 		multiprocessing multiprocessing/dummy \
 		pathlib \
 		profile \

@@ -141,6 +141,8 @@ PYTHONPATH=$(COREPYTHONPATH)
 #_interpreters _interpretersmodule.c
 #_json _json.c
 #_lsprof _lsprof.c rotatingtree.c
+#_math mathmodule.c
+#_math_integer mathintegermodule.c
 #_multiprocessing -I$(srcdir)/Modules/_multiprocessing _multiprocessing/multiprocessing.c _multiprocessing/semaphore.c
 #_opcode _opcode.c
 #_pickle _pickle.c
@@ -149,17 +151,15 @@ PYTHONPATH=$(COREPYTHONPATH)
 #_socket socketmodule.c
 #_statistics _statisticsmodule.c
 #_struct _struct.c
+#_sysconfig _sysconfig.c
 #_types _typesmodule.c
 #_typing _typingmodule.c
 #_zoneinfo _zoneinfo.c
 #array arraymodule.c
 #binascii binascii.c
 #cmath cmathmodule.c
-#math mathmodule.c
-#_math_integer mathintegermodule.c
 #mmap mmapmodule.c
 #select selectmodule.c
-#_sysconfig _sysconfig.c
 
 # XML
 #_elementtree _elementtree.c

@@ -52,8 +52,8 @@
 @MODULE__ZONEINFO_TRUE@_zoneinfo _zoneinfo.c
 
 # needs libm
-@MODULE_MATH_TRUE@math mathmodule.c
 @MODULE_CMATH_TRUE@cmath cmathmodule.c
+@MODULE__MATH_TRUE@_math mathmodule.c
 @MODULE__STATISTICS_TRUE@_statistics _statisticsmodule.c
 
 # _decimal uses libmpdec

diff --git a/Modules/mathintegermodule.c b/Modules/mathintegermodule.c
@@ -1236,39 +1236,7 @@ static PyMethodDef math_integer_methods[] = {
     {NULL,              NULL}           /* sentinel */
 };
 
-static int
-math_integer_exec(PyObject *module)
-{
-    /* Fix the __name__ attribute of the module and the __module__ attribute
-     * of its functions.
-     */
-    PyObject *name = PyUnicode_FromString("math.integer");
-    if (name == NULL) {
-        return -1;
-    }
-    if (PyObject_SetAttrString(module, "__name__", name) < 0) {
-        Py_DECREF(name);
-        return -1;
-    }
-    for (const PyMethodDef *m = math_integer_methods; m->ml_name; m++) {
-        PyObject *obj = PyObject_GetAttrString(module, m->ml_name);
-        if (obj == NULL) {
-            Py_DECREF(name);
-            return -1;
-        }
-        if (PyObject_SetAttrString(obj, "__module__", name) < 0) {
-            Py_DECREF(name);
-            Py_DECREF(obj);
-            return -1;
-        }
-        Py_DECREF(obj);
-    }
-    Py_DECREF(name);
-    return 0;
-}
-
 static PyModuleDef_Slot math_integer_slots[] = {
-    {Py_mod_exec, math_integer_exec},
     {Py_mod_multiple_interpreters, Py_MOD_PER_INTERPRETER_GIL_SUPPORTED},
     {Py_mod_gil, Py_MOD_GIL_NOT_USED},
     {0, NULL}

diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c
@@ -2976,7 +2976,6 @@ math_ulp_impl(PyObject *module, double x)
 static int
 math_exec(PyObject *module)
 {
-
     if (PyModule_Add(module, "pi", PyFloat_FromDouble(Py_MATH_PI)) < 0) {
         return -1;
     }
@@ -2993,32 +2992,6 @@ math_exec(PyObject *module)
     if (PyModule_Add(module, "nan", PyFloat_FromDouble(fabs(Py_NAN))) < 0) {
         return -1;
     }
-
-    PyObject *intmath = PyImport_ImportModule("_math_integer");
-    if (!intmath) {
-        return -1;
-    }
-#define IMPORT_FROM_INTMATH(NAME) do {                          \
-        if (PyModule_Add(module, #NAME,                         \
-                PyObject_GetAttrString(intmath, #NAME)) < 0) {  \
-            Py_DECREF(intmath);                                 \
-            return -1;                                          \
-        }                                                       \
-    } while(0)
-
-    IMPORT_FROM_INTMATH(comb);
-    IMPORT_FROM_INTMATH(factorial);
-    IMPORT_FROM_INTMATH(gcd);
-    IMPORT_FROM_INTMATH(isqrt);
-    IMPORT_FROM_INTMATH(lcm);
-    IMPORT_FROM_INTMATH(perm);
-    if (_PyImport_SetModuleString("math.integer", intmath) < 0) {
-        Py_DECREF(intmath);
-        return -1;
-    }
-    if (PyModule_Add(module, "integer", intmath) < 0) {
-        return -1;
-    }
     return 0;
 }
 
@@ -3095,15 +3068,15 @@ PyDoc_STRVAR(module_doc,
 
 static struct PyModuleDef mathmodule = {
     PyModuleDef_HEAD_INIT,
-    .m_name = "math",
+    .m_name = "_math",
     .m_doc = module_doc,
     .m_size = 0,
     .m_methods = math_methods,
     .m_slots = math_slots,
 };
 
 PyMODINIT_FUNC
-PyInit_math(void)
+PyInit__math(void)
 {
     return PyModuleDef_Init(&mathmodule);
 }
@@ -14,7 +14,7 @@ extern PyObject* PyInit_faulthandler(void);
 extern PyObject* PyInit__tracemalloc(void);
 extern PyObject* PyInit_gc(void);
 extern PyObject* PyInit__math_integer(void);
-extern PyObject* PyInit_math(void);
+extern PyObject* PyInit__math(void);
 extern PyObject* PyInit_nt(void);
 extern PyObject* PyInit__operator(void);
 extern PyObject* PyInit__signal(void);
@@ -102,7 +102,7 @@ struct _inittab _PyImport_Inittab[] = {
     {"faulthandler", PyInit_faulthandler},
     {"gc", PyInit_gc},
     {"_math_integer", PyInit__math_integer},
-    {"math", PyInit_math},
+    {"_math", PyInit__math},
     {"nt", PyInit_nt}, /* Use the NT os functions, not posix */
     {"_operator", PyInit__operator},
     {"_signal", PyInit__signal},

diff --git a/Python/stdlib_module_names.h b/Python/stdlib_module_names.h
@@ -7911,9 +7911,9 @@ PY_STDLIB_MOD([_posixshmem],
   [$POSIXSHMEM_CFLAGS], [$POSIXSHMEM_LIBS])
 
 dnl needs libm
+PY_STDLIB_MOD_SIMPLE([_math], [], [$LIBM])
 PY_STDLIB_MOD_SIMPLE([_statistics], [], [$LIBM])
 PY_STDLIB_MOD_SIMPLE([cmath], [], [$LIBM])
-PY_STDLIB_MOD_SIMPLE([math], [], [$LIBM])
 
 dnl needs libm and on some platforms librt
 PY_STDLIB_MOD_SIMPLE([_datetime], [], [$TIMEMODULE_LIB $LIBM])