diff --git a/mlir/lib/Dialect/Index/IR/IndexOps.cpp b/mlir/lib/Dialect/Index/IR/IndexOps.cpp
--- a/mlir/lib/Dialect/Index/IR/IndexOps.cpp
+++ b/mlir/lib/Dialect/Index/IR/IndexOps.cpp
@@ -142,9 +142,22 @@
 //===----------------------------------------------------------------------===//
 
 OpFoldResult MulOp::fold(FoldAdaptor adaptor) {
-  return foldBinaryOpUnchecked(
-      adaptor.getOperands(),
-      [](const APInt &lhs, const APInt &rhs) { return lhs * rhs; });
+  if (OpFoldResult result = foldBinaryOpUnchecked(
+          adaptor.getOperands(),
+          [](const APInt &lhs, const APInt &rhs) { return lhs * rhs; });
+      !result.isNull())
+    return result;
+
+  if (auto rhs = dyn_cast_or_null<IntegerAttr>(adaptor.getRhs())) {
+    // Fold `mul(x, 1) -> x`.
+    if (rhs.getValue().isOne())
+      return getLhs();
+    // Fold `mul(x, 0) -> 0`.
+    if (rhs.getValue().isZero())
+      return rhs;
+  }
+
+  return {};
 }
 
 //===----------------------------------------------------------------------===//
diff --git a/mlir/test/Dialect/Index/index-canonicalize.mlir b/mlir/test/Dialect/Index/index-canonicalize.mlir
--- a/mlir/test/Dialect/Index/index-canonicalize.mlir
+++ b/mlir/test/Dialect/Index/index-canonicalize.mlir
@@ -521,3 +521,13 @@
   // CHECK: return %true, %false
   return %1, %2 : i1, i1
 }
+
+// CHECK-LABEL: @mul_identity
+func.func @mul_identity(%arg0: index) -> (index, index) {
+  %idx0 = index.constant 0
+  %idx1 = index.constant 1
+  %0 = index.mul %arg0, %idx0
+  %1 = index.mul %arg0, %idx1
+  // CHECK: return %idx0, %arg0
+  return %0, %1 : index, index
+}