Trac #34261: Allow multiplication of a left and right noncommutative …

…ideal

Their product has a simple generating set, but right now Sage does not
allow this:
{{{
sage: MS = MatrixSpace(QQ,2,2)
sage: IL = MS * [2*MS.0,3*MS.1]
sage: IR = MS.3 * MS
sage: IL * IR
...
NotImplementedError: Cannot multiply non-commutative ideals.
}}}
We fix this.

URL: https://trac.sagemath.org/34261
Reported by: tscrim
Ticket author(s): Travis Scrimshaw
Reviewer(s): Dave Morris

src/sage/rings/noncommutative_ideals.pyx

@@ -337,14 +337,26 @@ class Ideal_nc(Ideal_generic):
         return self.__side
 
     def __mul__(self, other):
-        """
+        r"""
+        Multiply ``self`` with ``other``.
+
         Multiplication of a one-sided ideal with its ring from the other side
         yields a two-sided ideal.
 
+        Let `L` (resp. `R`) be a left (resp. right) ideal, then the product
+        `LR` is a twosided ideal generated by `x y`, where `x` (resp. `y`)
+        is a generator of `L` (resp. `R`).
+
+        .. TODO::
+
+            The product of left (resp. right) ideals is a left
+            (resp. right) ideal. However, these do not necessarily have
+            simple generating sets.
+
         TESTS::
 
             sage: MS = MatrixSpace(QQ,2,2)
-            sage: IL = MS*[2*MS.0,3*MS.1]; IL
+            sage: IL = MS * [2*MS.0,3*MS.1]; IL
             Left Ideal
             (
               [2 0]
@@ -361,7 +373,7 @@ class Ideal_nc(Ideal_generic):
               [0 1]
             )
              of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
-            sage: IL*MS     # indirect doctest
+            sage: IL * MS
             Twosided Ideal
             (
               [2 0]
@@ -371,12 +383,26 @@ class Ideal_nc(Ideal_generic):
               [0 0]
             )
              of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
-            sage: IR*IR
+
+            sage: IL * IR
+            Twosided Ideal 
+            (
+              [0 3]
+              [0 0]
+            )
+             of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
+
+            sage: IR * IR
             Traceback (most recent call last):
             ...
-            NotImplementedError: Cannot multiply non-commutative ideals.
-
+            NotImplementedError: cannot multiply non-commutative ideals
         """
+        if isinstance(other, Ideal_nc) and self.ring() == other.ring():
+            if self.side() == "left" and other.side() == "right":
+                gens = [z for z in (x * y for x in self.gens() for y in other.gens()) if z]
+                return self.ring().ideal(gens, side='twosided')
+            raise NotImplementedError("cannot multiply non-commutative ideals")
+
         if not isinstance(other, Ideal_nc):
             # Perhaps other is a ring and thus has its own
             # multiplication.
@@ -390,4 +416,6 @@ class Ideal_nc(Ideal_generic):
                 if other.side() == 'left':
                     return other
                 return other.ring().ideal(other.gens(), side='twosided')
-        raise NotImplementedError("Cannot multiply non-commutative ideals.")
+
+        raise NotImplementedError("cannot multiply non-commutative ideals")
+