* algebra/catdef.spad.pamphlet (LinearlyExplicitRingOver): Now

	extends LeftModule R.
	[reducedSystem: Vector % -> Matrix R]: New overload.
	* algebra/lindep.spad.pamphlet (LinearDependence): Additionally
	require Ring on the second parameter.
	(IntegerLinearDependence): Add similar requirement on the parameter.

1 files changed, 8 insertions, 3 deletions

diff --git a/src/algebra/catdef.spad.pamphlet b/src/algebra/catdef.spad.pamphlet
index 4c6317f3..0a1a435b 100644
--- a/src/algebra/catdef.spad.pamphlet
+++ b/src/algebra/catdef.spad.pamphlet
@@ -1013,7 +1013,7 @@ LeftModule(R:Rng):Category == Join(AbelianGroup, LeftLinearSet R)
 )abbrev category LINEXP LinearlyExplicitRingOver
 ++ Author:
 ++ Date Created:
-++ Date Last Updated:
+++ Date Last Updated: June 14, 2010
 ++ Basic Functions:
 ++ Related Constructors:
 ++ Also See:
@@ -1021,8 +1021,13 @@ LeftModule(R:Rng):Category == Join(AbelianGroup, LeftLinearSet R)
 ++ Keywords:
 ++ References:
 ++ Description:
-++ An extension ring with an explicit linear dependence test.
-LinearlyExplicitRingOver(R:Ring): Category == Ring with
+++    An extension of left-module with an explicit linear dependence test.
+LinearlyExplicitRingOver(R:Ring): Category == LeftModule R with
+  reducedSystem: Vector % -> Matrix R
+    ++ \spad{reducedSystem [v1,...,vn]} returns a matrix \spad{M}
+    ++ with coefficients in \spad{R} such that the system of equations
+    ++ \spad{c1*v1 + ... + cn*vn = 0$%} has the same solution as
+    ++ \spad{c * M = 0} where \spad{c} is the row vector \spad{[c1,...cn]}.
   reducedSystem: Matrix % -> Matrix R
     ++ reducedSystem(A) returns a matrix B such that \spad{A x = 0} and \spad{B x = 0}
     ++ have the same solutions in R.