diff --git a/doc/source/whatsnew/v1.4.0.rst b/doc/source/whatsnew/v1.4.0.rst
index 4914d213accac..32f99c7852cca 100644
--- a/doc/source/whatsnew/v1.4.0.rst
+++ b/doc/source/whatsnew/v1.4.0.rst
@@ -411,6 +411,7 @@ Numeric
 - Bug in :meth:`DataFrame.rank` treating missing values and extreme values as equal (for example ``np.nan`` and ``np.inf``), causing incorrect results when ``na_option="bottom"`` or ``na_option="top`` used (:issue:`41931`)
 - Bug in ``numexpr`` engine still being used when the option ``compute.use_numexpr`` is set to ``False`` (:issue:`32556`)
 - Bug in :class:`DataFrame` arithmetic ops with a subclass whose :meth:`_constructor` attribute is a callable other than the subclass itself (:issue:`43201`)
+- Bug in arithmetic operations involving :class:`RangeIndex` where the result would have the incorrect ``name`` (:issue:`43962`)
 -
 
 Conversion
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 4003165a7ddc6..d219361dbdd57 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -885,9 +885,8 @@ def _arith_method(self, other, op):
             step = op
 
         # TODO: if other is a RangeIndex we may have more efficient options
-        other = extract_array(other, extract_numpy=True, extract_range=True)
-
-        left, right = self, other
+        right = extract_array(other, extract_numpy=True, extract_range=True)
+        left = self
 
         try:
             # apply if we have an override
@@ -907,7 +906,8 @@ def _arith_method(self, other, op):
                 rstart = op(left.start, right)
                 rstop = op(left.stop, right)
 
-            result = type(self)(rstart, rstop, rstep, name=self.name)
+            res_name = ops.get_op_result_name(self, other)
+            result = type(self)(rstart, rstop, rstep, name=res_name)
 
             # for compat with numpy / Int64Index
             # even if we can represent as a RangeIndex, return
diff --git a/pandas/tests/arithmetic/test_numeric.py b/pandas/tests/arithmetic/test_numeric.py
index f06e5e9bdc93b..ec17b9f0bfd52 100644
--- a/pandas/tests/arithmetic/test_numeric.py
+++ b/pandas/tests/arithmetic/test_numeric.py
@@ -693,11 +693,10 @@ def test_mul_float_series(self, numeric_idx):
         tm.assert_series_equal(result, expected)
 
     def test_mul_index(self, numeric_idx):
-        # in general not true for RangeIndex
         idx = numeric_idx
-        if not isinstance(idx, RangeIndex):
-            result = idx * idx
-            tm.assert_index_equal(result, idx ** 2)
+
+        result = idx * idx
+        tm.assert_index_equal(result, idx ** 2)
 
     def test_mul_datelike_raises(self, numeric_idx):
         idx = numeric_idx
@@ -1090,11 +1089,11 @@ def test_ufunc_compat(self, holder):
         box = Series if holder is Series else Index
 
         if holder is RangeIndex:
-            idx = RangeIndex(0, 5)
+            idx = RangeIndex(0, 5, name="foo")
         else:
-            idx = holder(np.arange(5, dtype="int64"))
+            idx = holder(np.arange(5, dtype="int64"), name="foo")
         result = np.sin(idx)
-        expected = box(np.sin(np.arange(5, dtype="int64")))
+        expected = box(np.sin(np.arange(5, dtype="int64")), name="foo")
         tm.assert_equal(result, expected)
 
     @pytest.mark.parametrize("holder", [Int64Index, UInt64Index, Float64Index, Series])
@@ -1212,6 +1211,8 @@ class TestNumericArithmeticUnsorted:
     def check_binop(self, ops, scalars, idxs):
         for op in ops:
             for a, b in combinations(idxs, 2):
+                a = a._rename("foo")
+                b = b._rename("bar")
                 result = op(a, b)
                 expected = op(Int64Index(a), Int64Index(b))
                 tm.assert_index_equal(result, expected)