diff --git a/src/Model/Relations/BelongsTo.php b/src/Model/Relations/BelongsTo.php
index 67c6c6a..5ae3416 100755
--- a/src/Model/Relations/BelongsTo.php
+++ b/src/Model/Relations/BelongsTo.php
@@ -152,15 +152,23 @@ public function match(array $models, Collection $results, $relation)
         $dictionary = [];
 
         foreach ($results as $result) {
-            $dictionary[$result->getAttribute($owner)] = $result;
+            // The owner key can be null when not selected; null as an array offset is deprecated since PHP 8.1
+            $key = $result->getAttribute($owner);
+
+            if ($key !== null) {
+                $dictionary[$key] = $result;
+            }
         }
 
         // Once we have the dictionary constructed, we can loop through all the parents
         // and match back onto their children using these keys of the dictionary and
         // the primary key of the children to map them onto the correct instances.
         foreach ($models as $model) {
-            if (isset($dictionary[$model->{$foreign}])) {
-                $model->setRelation($relation, $dictionary[$model->{$foreign}]);
+            // The foreign key is nullable, and using null as an array offset is deprecated since PHP 8.1
+            $key = $model->{$foreign};
+
+            if ($key !== null && isset($dictionary[$key])) {
+                $model->setRelation($relation, $dictionary[$key]);
             }
         }
 
diff --git a/src/Model/Relations/MorphTo.php b/src/Model/Relations/MorphTo.php
index b569e26..1a2cae2 100644
--- a/src/Model/Relations/MorphTo.php
+++ b/src/Model/Relations/MorphTo.php
@@ -245,9 +245,15 @@ public function morphWith(array $with)
     protected function buildDictionary(Collection $models)
     {
         foreach ($models as $model) {
-            if ($model->{$this->morphType}) {
-                $this->dictionary[$model->{$this->morphType}][$model->{$this->foreignKey}][] = $model;
+            // Both are nullable, and using null as an array offset is deprecated since PHP 8.1
+            $type = $model->{$this->morphType};
+            $key = $model->{$this->foreignKey};
+
+            if (empty($type) || $key === null) {
+                continue;
             }
+
+            $this->dictionary[$type][$key][] = $model;
         }
     }