diff --git a/parso/python/tree.py b/parso/python/tree.py
index 7d4a490..3d5e842 100644
--- a/parso/python/tree.py
+++ b/parso/python/tree.py
@@ -210,15 +210,12 @@ class Name(_LeafWithoutNewlines):
         """
         Returns None if there's on definition for a name.
 
-        :param import_name_alway: Specifies if an import name is always a
+        :param import_name_always: Specifies if an import name is always a
             definition. Normally foo in `from foo import bar` is not a
             definition.
         """
         node = self.parent
         type_ = node.type
-        if type_ in ('power', 'atom_expr'):
-            # In `self.x = 3` self is not a definition, but x is.
-            return None
 
         if type_ in ('funcdef', 'classdef'):
             if self == node.name:
@@ -1037,6 +1034,14 @@ def _defined_names(current):
             trailer = current.children[-1]
             if trailer.children[0] == '.':
                 names.append(trailer.children[1])
+            elif trailer.children[0] == '[':
+                for node in current.children[-2::-1]:
+                    if node.type == 'trailer':
+                        names.append(node.children[1])
+                        break
+                    if node.type == 'name':
+                        names.append(node)
+                        break
     else:
         names.append(current)
     return names
diff --git a/test/test_parser_tree.py b/test/test_parser_tree.py
index 69cfe76..d9d5893 100644
--- a/test/test_parser_tree.py
+++ b/test/test_parser_tree.py
@@ -180,3 +180,42 @@ def top_function_three():
 
     r = get_raise_stmts(code, 2) #  Lists inside try-catch
     assert len(list(r)) == 2
+
+
+@pytest.mark.parametrize(
+    'code, name_index, is_definition', [
+        ('x = 3', 0, True),
+        ('x.y = 3', 0, False),
+        ('x.y = 3', 1, True),
+        ('x.y = u.v = z', 0, False),
+        ('x.y = u.v = z', 1, True),
+        ('x.y = u.v = z', 2, False),
+        ('x.y = u.v, w = z', 3, True),
+        ('x.y = u.v, w = z', 4, True),
+        ('x.y = u.v, w = z', 5, False),
+
+        ('x, y = z', 0, True),
+        ('x, y = z', 1, True),
+        ('x, y = z', 2, False),
+        ('x, y = z', 2, False),
+        ('x[0], y = z', 2, False),
+        ('x[0] = z', 0, True),
+        ('x[0], y = z', 0, True),
+        ('x: int = z', 0, True),
+        ('x: int = z', 1, False),
+        ('x: int = z', 2, False),
+        ('x: int', 0, True),
+        ('x: int', 1, False),
+    ]
+)
+def test_is_definition(code, name_index, is_definition):
+    module = parse(code, version='3.8')
+    name = module.get_first_leaf()
+    while True:
+        if name.type == 'name':
+            if name_index == 0:
+                break
+            name_index -= 1
+        name = name.get_next_leaf()
+
+    assert name.is_definition() == is_definition