The following functions allow simple access and operations on module elements. Some of them use the module structure and refer to the column structure of an element; others use the polynomial structure and ignore the column structure.

Subsections

• Access
• Arithmetic
• Predicates
• Other Operations

Given an element f of the module M over P and of degree r, return the sequence [f_1, ..., f_r] of r elements from P corresponding to f.

Given an element f of the module M over P and of degree r, together with an integer i in the range [1 .. r], return the i-th component of f as an element of P.

Given elements f and g of the module M, return the sum f + g of f and g. Note that the result is always reduced to the unique normal form modulo the quotient relations of M.

Given elements f and g of the module M, return the difference f - g of f and g. Note that the result is always reduced to the unique normal form modulo the quotient relations of M.

- f : ModMPolElt -> ModMPolElt

Given an element f of the module M, return the negation -f of f.

Given a scalar ring element s and an element f of the module M, such that s is coercible into the coefficient ring of M, return the element sf.

f * s : ModMPolElt, RngMPolElt -> ModMPolElt

Given a scalar ring element s and an element f of the module M, such that s is coercible into the coefficient ring of M, return the element fs.

Given a scalar ring element s and an element f of the module M, such that s is coercible into the coefficient ring of M and s divides all components of f, return the quotient of f by s.

(Procedure.) Given a scalar ring element s and an element f of the module M, such that s is coercible into the coefficient ring of M and s divides all components of f, replace f by the quotient of f by s.

Given elements f and g of the module M such that the leading monomials of f and g have the same column, return the S-polynomial of f and g. Note that the result is always reduced to the unique normal form modulo the quotient relations of M.

Given an element f of the module M, return whether f is the zero element of M.

Given elements f and g of the module M, return whether f and g are equal.

Given an element f of a module S together with a compatible module M, return whether f is in M.

Given an element f of the module M, return the normalized form of f (so that the leading monomial of f is monic).

Given an element f of the module S, together with a compatible module M, return the normal form of f with respect to M.

Given an element f of the module S, together with a compatible module M such that f is in M, return the coordinates of f with respect to the basis of M (whose components lie in the coefficient ring of M).

> P<x, y, z> := PolynomialRing(RationalField(), 3);
> M := Module(P, 3);
> a := M ! [1, x, y];
> b := M ! [x, y, z]; 
> a;
(1 x y)
> b;
(x y z)
> a + b;
(x + 1 x + y y + z)
> z*a;
(  z x*z y*z)
> Eltseq(z * a + b);
[
    x + z,
    x*z + y,
    y*z + z
]