Singular Manual: ringlist (plural)

7.3.24 ringlist (plural)

Syntax:

ringlist ( ring_expression )
ringlist ( qring_expression )

Type:

list

Purpose:

decomposes a ring/qring into a list of 6 (or 4 in the commutative case) components.
The first 4 components are common both for the commutative and for the non-commutative cases, the 5th and the 6th appear only in the non-commutative case.

upper triangle square matrix with nonzero upper triangle, containing structural coefficients of a G-algebra (this corresponds to the matrix C from the definition of G-algebras)
square matrix, containing structural polynomials of a G-algebra (this corresponds to the matrix D from the definition of G-algebras)

Note:

After modifying a list acquired with ringlist, one can construct a corresponding ring with ring(list).

Example:

// consider the quantized Weyl algebra
ring r = (0,q),(x,d),Dp;
def RS=nc_algebra(q,1);
setring RS; RS;
==> // coefficients: QQ(q)
==> // number of vars : 2
==> //        block   1 : ordering Dp
==> //                  : names    x d
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    dx=(q)*xd+1
list l = ringlist(RS);
l;
==> [1]:
==>    [1]:
==>       0
==>    [2]:
==>       [1]:
==>          q
==>    [3]:
==>       [1]:
==>          [1]:
==>             lp
==>          [2]:
==>             1
==>    [4]:
==>       _[1]=0
==> [2]:
==>    [1]:
==>       x
==>    [2]:
==>       d
==> [3]:
==>    [1]:
==>       [1]:
==>          Dp
==>       [2]:
==>          1,1
==>    [2]:
==>       [1]:
==>          C
==>       [2]:
==>          0
==> [4]:
==>    _[1]=0
==> [5]:
==>    _[1,1]=0
==>    _[1,2]=(q)
==>    _[2,1]=0
==>    _[2,2]=0
==> [6]:
==>    _[1,1]=0
==>    _[1,2]=1
==>    _[2,1]=0
==>    _[2,2]=0
// now, change the relation d*x = q*x*d +1
// into the relation d*x=(q2+1)*x*d + q*d + 1
matrix S = l[5]; // matrix of coefficients
S[1,2] = q^2+1;
l[5] = S;
matrix T = l[6]; // matrix of polynomials
T[1,2] = q*d+1;
l[6] = T;
def rr = ring(l);
setring rr; rr;
==> // coefficients: QQ(q)
==> // number of vars : 2
==> //        block   1 : ordering Dp
==> //                  : names    x d
==> //        block   2 : ordering C
==> // noncommutative relations:
==> //    dx=(q2+1)*xd+(q)*d+1