Singular Manual: ring operations (plural)

Forward: ring related functions (plural)

7.2.7.2 ring operations (plural)

+

construct a tensor product $C = A\otimes_{\bf {K} } B$ of two

-algebras

and

over the ground field. Let, e.g.,

$A= k_1 \langle x_1, \ldots ,x_n \mid$ $\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}\}, 1 \leq i <j \leq n \rangle$ , and $B= k_2 \langle y_1, \ldots ,y_m \mid$ $\{ y_j y_i=q_{ij} \cdot y_i y_j + r_{ij}\}, 1 \leq i <j \leq m \rangle$

be two -algebras, then is defined to be the algebra

$C = K \langle x_1, \ldots ,x_n, y_1, \ldots ,y_m \mid$ $\{ x_j x_i=c_{ij} \cdot x_i x_j + d_{ij}, 1 \leq i <j \leq n\}$ , $\{ y_j y_i=q_{ij} \cdot y_i y_j + r_{ij}, 1 \leq i <j \leq m\}$ , $\{ y_j x_i = x_i y_j, 1 \leq j \leq m, 1 \leq i \leq n\} \rangle$ .

Concerning the ground fields resp. of resp. , take the following guidelines for $A\otimes_{\bf {K} } B$ into consideration:

Neither nor may be or .
If the characteristic of and differs, then one of them must be .
At most one of and may have parameters.
If one of and is an algebraic extension of it may not be defined by a charstr of type (p^n,a).

One can create a ring using ring(list), see also ringlist.

Example:

LIB "ncalg.lib"; def a = makeUsl2(); // U(sl_2) in e,f,h presentation ring W0 = 0,(x,d),dp; def W = Weyl(); // 1st Weyl algebra in x,d def S = a+W; setring S; S; ==> // characteristic : 0 ==> // number of vars : 5 ==> // block 1 : ordering dp ==> // : names e f h ==> // block 2 : ordering dp ==> // : names x d ==> // block 3 : ordering C ==> // noncommutative relations: ==> // fe=ef-h ==> // he=eh+2e ==> // hf=fh-2f ==> // dx=xd+1

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