# Singular

### A.3.10 Normalization

The normalization will be computed for a reduced ring . The result is a list of rings; ideals are always called `norid` in the rings of this list. The normalization of is the product of the factor rings of the rings in the list divided out by the ideals `norid`.

 ``` LIB "normal.lib"; // ----- first example: rational quadruple point ----- ring R=32003,(x,y,z),wp(3,5,15); ideal I=z*(y3-x5)+x10; list pr=normal(I); ==> ==> // 'normal' created a list, say nor, of two elements. ==> // To see the list type ==> nor; ==> ==> // * nor is a list of 1 ring(s). ==> // To access the i-th ring nor[i], give it a name, say Ri, and type ==> def R1 = nor; setring R1; norid; normap; ==> // For the other rings type first (if R is the name of your base ring) ==> setring R; ==> // and then continue as for R1. ==> // Ri/norid is the affine algebra of the normalization of R/P_i where ==> // P_i is the i-th component of a decomposition of the input ideal id ==> // and normap the normalization map from R to Ri/norid. ==> ==> // * nor is a list of 1 ideal(s). Let ci be the last generator ==> // of the ideal nor[i]. Then the integral closure of R/P_i is ==> // generated as R-submodule of the total ring of fractions by ==> // 1/ci * nor[i]. def S=pr; setring S; norid; ==> norid=T(2)*x+y*z ==> norid=T(1)*x^2-T(2)*y ==> norid=-T(1)*y+x^7-x^2*z ==> norid=T(1)*y^2*z+T(2)*x^8-T(2)*x^3*z ==> norid=T(1)^2+T(2)*z+x^4*y*z ==> norid=T(1)*T(2)+x^6*z-x*z^2 ==> norid=T(2)^2+T(1)*x*z ==> norid=x^10-x^5*z+y^3*z // ----- second example: union of straight lines ----- ring R1=0,(x,y,z),dp; ideal I=(x-y)*(x-z)*(y-z); list qr=normal(I); ==> ==> // 'normal' created a list, say nor, of two elements. ==> // To see the list type ==> nor; ==> ==> // * nor is a list of 2 ring(s). ==> // To access the i-th ring nor[i], give it a name, say Ri, and type ==> def R1 = nor; setring R1; norid; normap; ==> // For the other rings type first (if R is the name of your base ring) ==> setring R; ==> // and then continue as for R1. ==> // Ri/norid is the affine algebra of the normalization of R/P_i where ==> // P_i is the i-th component of a decomposition of the input ideal id ==> // and normap the normalization map from R to Ri/norid. ==> ==> // * nor is a list of 2 ideal(s). Let ci be the last generator ==> // of the ideal nor[i]. Then the integral closure of R/P_i is ==> // generated as R-submodule of the total ring of fractions by ==> // 1/ci * nor[i]. def S1=qr; def S2=qr; setring S1; norid; ==> norid=-T(1)*y+T(1)*z+x-z ==> norid=T(1)*x-T(1)*y ==> norid=T(1)^2-T(1) ==> norid=x^2-x*y-x*z+y*z setring S2; norid; ==> norid=y-z ```

### Misc 