diff --git a/jax_scaled_arithmetics/core/interpreters.py b/jax_scaled_arithmetics/core/interpreters.py
index 91ef749..6e46c13 100644
--- a/jax_scaled_arithmetics/core/interpreters.py
+++ b/jax_scaled_arithmetics/core/interpreters.py
@@ -113,7 +113,9 @@ def to_scaled_array(val):
 
         # Primitive is supported by `autoscale`?
         if eqn.primitive not in _scaled_ops_registry:
-            raise NotImplementedError(f"{eqn.primitive} does not have an implementation for ScaledArray inputs yet")
+            raise NotImplementedError(
+                f"'{eqn.primitive}' JAX primitive does not have an implementation for ScaledArray inputs yet."
+            )
         outvals = _scaled_ops_registry[eqn.primitive](*invals, **eqn.params)
         if not eqn.primitive.multiple_results:
             outvals = [outvals]
diff --git a/jax_scaled_arithmetics/lax/__init__.py b/jax_scaled_arithmetics/lax/__init__.py
index 65b52cc..4d7e612 100644
--- a/jax_scaled_arithmetics/lax/__init__.py
+++ b/jax_scaled_arithmetics/lax/__init__.py
@@ -1,2 +1,3 @@
 # Copyright (c) 2023 Graphcore Ltd. All rights reserved.
+from .base_scaling_primitives import set_scaling, set_scaling_p  # noqa: F401
 from .scaled_ops import *  # noqa: F401, F403
diff --git a/jax_scaled_arithmetics/lax/base_scaling_primitives.py b/jax_scaled_arithmetics/lax/base_scaling_primitives.py
new file mode 100644
index 0000000..5d29715
--- /dev/null
+++ b/jax_scaled_arithmetics/lax/base_scaling_primitives.py
@@ -0,0 +1,67 @@
+# Copyright (c) 2023 Graphcore Ltd. All rights reserved.
+from typing import Sequence, Union
+
+import jax
+from jax import core
+from jax.interpreters import mlir
+from jax.interpreters.mlir import LoweringRuleContext, ir
+
+from jax_scaled_arithmetics.core import ScaledArray, register_scaled_op
+
+set_scaling_p = core.Primitive("set_scaling_p")
+"""`set_scaling` JAX primitive.
+
+In standard JAX, this is just an identity operation, ignoring the `scale`
+input, just returning unchanged the `data` component.
+
+In JAX Scaled Arithmetics mode, it will rebalance the data term to
+return a ScaledArray semantically equivalent.
+"""
+
+
+def set_scaling(data: jax.Array, scale: jax.Array) -> jax.Array:
+    return set_scaling_p.bind(data, scale)
+
+
+def set_scaling_impl(data: jax.Array, scale: jax.Array) -> jax.Array:
+    return data
+
+
+def set_scaling_abstract_eval(data: core.ShapedArray, scale: core.ShapedArray) -> core.ShapedArray:
+    return data
+
+
+def set_default_mlir_lowering(
+    ctx: LoweringRuleContext, *args: Union[ir.Value, Sequence[ir.Value]]
+) -> Sequence[Union[ir.Value, Sequence[ir.Value]]]:
+    # Just forwarding `data` term, ignoring the scaling.
+    return (args[0],)
+
+
+def scaled_set_scaling(val: ScaledArray, scale: ScaledArray) -> ScaledArray:
+    """Scaled `set_scaling` implementation: rebalancing the data using the new scale value."""
+    assert isinstance(val, ScaledArray)
+    assert isinstance(scale, ScaledArray)
+    assert scale.shape == ()
+
+    scale_value = scale.to_array()
+    # Rebalancing data term.
+    data = val.data * (val.scale / scale_value)
+    return ScaledArray(data, scale_value)
+
+
+set_scaling_p.multiple_results = False
+# Register the abstract evaluation with JAX
+set_scaling_p.def_abstract_eval(set_scaling_abstract_eval)
+# Register the primal implementation with JAX
+set_scaling_p.def_impl(set_scaling_impl)
+# Register MLIR lowering.
+mlir.register_lowering(set_scaling_p, set_default_mlir_lowering)
+register_scaled_op(set_scaling_p, scaled_set_scaling)
+
+
+stop_scaling_p = core.Primitive("stop_scaling_p")
+"""`stop_scaling` JAX primitive.
+
+Similar in principle to `jax.lax.stop_gradient`
+"""
diff --git a/tests/lax/test_base_scaling_primitives.py b/tests/lax/test_base_scaling_primitives.py
new file mode 100644
index 0000000..7b7e697
--- /dev/null
+++ b/tests/lax/test_base_scaling_primitives.py
@@ -0,0 +1,36 @@
+# Copyright (c) 2023 Graphcore Ltd. All rights reserved.
+import chex
+import jax.numpy as jnp
+import numpy as np
+import numpy.testing as npt
+
+from jax_scaled_arithmetics.core import ScaledArray, autoscale, scaled_array
+from jax_scaled_arithmetics.lax import set_scaling
+
+
+class SetScalingPrimitiveTests(chex.TestCase):
+    @chex.variants(with_jit=True, without_jit=True)
+    def test__set_scaling_primitive__proper_result_without_autoscale(self):
+        def fn(arr, scale):
+            return set_scaling(arr, scale)
+
+        fn = self.variant(fn)
+        arr = jnp.array([2, 3], dtype=np.float32)
+        scale = jnp.array(4, dtype=np.float32)
+        out = fn(arr, scale)
+        npt.assert_array_equal(out, arr)
+
+    @chex.variants(with_jit=True, without_jit=False)
+    def test__set_scaling_primitive__proper_result_with_autoscale(self):
+        def fn(arr, scale):
+            return set_scaling(arr, scale)
+
+        fn = self.variant(autoscale(fn))
+        arr = scaled_array([-1.0, 2.0], 1.0, dtype=np.float32)
+        # TODO: support scalar here!
+        scale = scaled_array(1.0, 4.0, dtype=np.float32)
+        out = fn(arr, scale)
+        # Unchanged output tensor!
+        assert isinstance(out, ScaledArray)
+        npt.assert_array_equal(out.scale, scale)
+        npt.assert_array_equal(out, arr)