5.1.116 preimage

Syntax:

preimage ( map )
preimage ( ring_name, map_name, ideal_name )
preimage ( ring_name, ideal_expression, ideal_name )

Type:

ring
ideal

Purpose:

returns the source ring of a map (in the first case) or returns the preimage of an ideal under a given map.
The second argument has to be a map from the basering to the given ring (or an ideal defining such a map), and the ideal has to be an ideal in the given ring.

Note:

As preimage is handling ideals (not polynomials), the result of a preimage calculation of a principal ideal is (the closure of) the preimage of the ideal, not that of the polynomial.

Example:

ring r1=32003,(x,y,z,w),lp; ring r=32003,(x,y,z),dp; ideal i=x,y,z; ideal i1=x,y; ideal i0=0; map f=r1,i; nameof (preimage (f)); ==> r1 setring r1; ideal i1=preimage(r,f,i1); i1; ==> i1[1]=w ==> i1[2]=y ==> i1[3]=x // the kernel of f preimage(r,f,i0); ==> _[1]=w // or, use: kernel(r,f); ==> _[1]=w