Trac #31178: inverse_image of element fails for morphism of quotient rings

Reported in [https://ask.sagemath.org/question/55099/inverse-image-
under-ring-homomorphism/ Ask SageMath question #55099]:

{{{
sage: R.<x, y> = QQ[]
sage: R1 = R.quotient(R.ideal(y^2 - x^3))
sage: R2 = R.quotient(R.ideal(x))
sage: h = R1.hom([R2(y^2), R2(y^3)], R2)
sage: h(R1(x))
ybar^2
sage: h.inverse_image(R2(y^2))
...
ValueError: element ybar^2 does not have preimage
}}}

URL: https://trac.sagemath.org/31178
Reported by: rburing
Ticket author(s): Markus Wageringel
Reviewer(s): Ricardo Buring

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=4913bd4c0630ad8756ddeb26a2efa33b6295b23f
-md5=6846b5857be492f8090bc00da88d9943
-cksum=223562041
+sha1=52f6e81ca1a215877ab83b5621d61ed93a428b11
+md5=cbb4222eb79ecc5637a81b3d57be6ca7
+cksum=2870898122

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-f14a482e2d5c81ecdc938ac65a5c24ddb9d65bc1
+3c618df00987db8f541521227d669f4d141fac27

src/sage/rings/morphism.pyx

@@ -1053,11 +1053,30 @@ cdef class RingHomomorphism(RingMap):
             sage: f = R.hom([y, x], R)
             sage: f.inverse_image(x), f.inverse_image(y)  # indirect doctest
             (0, 0)
+
+        Check cases involving quotient rings in which a generator is constant
+        (:trac:`31178`)::
+
+            sage: R.<x,y> = QQ[]
+            sage: B.<c,d> = R.quotient(R.ideal(x))
+            sage: g = R.hom([d^2, d^3], B)
+            sage: g.inverse_image(d)
+            Traceback (most recent call last):
+            ...
+            ValueError: element d does not have preimage
+            sage: g.inverse_image(d^2)
+            x
+            sage: g.inverse_image(d^3)
+            y
+            sage: A.<a,b> = R.quotient(R.ideal(y^2 - x^3))
+            sage: h = A.hom([d^2, d^3], B)
+            sage: h.inverse_image(d^2)
+            a
         """
         graph, from_B, to_A = self._graph_ideal()
-        gens_B = graph.ring().gens()[:self.codomain().ngens()]
+        gens_A = graph.ring().gens()[-self.domain().ngens():]
         a = graph.reduce(from_B(b))
-        if not (a.lm() < min(gens_B)) and not a.is_zero():
+        if not all(x in gens_A for x in a.lm().variables()):
             raise ValueError(f"element {b} does not have preimage")
         return to_A(a)