### 7.3.8 imap (plural)

`Syntax:`
  `imap (` ring_name`,` name `)`
`Type:`
  number, poly, vector, ideal, module, matrix or list (the same type as the second argument)
`Purpose:`
  identity map on common subrings. `imap` is the map between rings and qrings with compatible ground fields which is the identity on variables and parameters of the same name and 0 otherwise. (See map (plural) for a description of possible mappings between different ground fields). Useful for mappings from a homogenized ring to the original ring or for mappings from/to rings with/without parameters. Compared with `fetch`, `imap` uses the names of variables and parameters. Unlike `map` and `fetch`, `imap` can map parameters to variables.
`Example:`

      LIB "ncalg.lib";
      ring ABP=0,(p4,p5,a,b),dp; //  a commutative ring
      def Usl3 = makeUsl(3);
      def BIG  = Usl3+ABP;
      setring BIG;
      poly P4 = 3*x(1)*y(1)+3*x(2)*y(2)+3*x(3)*y(3);
      P4 = P4 +h(1)^2+h(1)*h(2)+h(2)^2-3*h(1)-3*h(2);
      // P4 is a central element of Usl3 of degree 2
      poly P5 = 4*x(1)*y(1) + h(1)^2 - 2*h(1);
      // P5 is a central element of the subalgebra of U(sl_3),
      // generated by x(1),y(1),h(1)
      ideal J = x(1),x(2),h(1)-a,h(2)-b;
      // we are interested in the module U(sl_3)/J,
      // which depends on parameters a,b
      ideal I = p4-P4, p5-P5;
      ideal K = I, J;
      ideal E = eliminate(K,x(1)*x(2)*x(3)*y(1)*y(2)*y(3)*h(1)*h(2));
      E; // this is the ideal of central characters in ABP
      ==> E[1]=a*b+b^2-p4+p5+a+3*b
      ==> E[2]=a^2-p5+2*a
      ==> E[3]=b^3+p4*a-p5*a-a^2-p4*b+3*b^2
      // what are the characters on nonzero a,b?
      ring abP = (0,a,b),(p4,p5),dp;
      ideal abE = imap(BIG, E);
      option(redSB);
      option(redTail);
      abE = std(abE);
      // here come characters (indeed, we have only one)
      // that is a maximal ideal in K[p4,p5]
      abE;
      ==> abE[1]=p5+(-a^2-2*a)
      ==> abE[2]=p4+(-a^2-a*b-3*a-b^2-3*b)

See fetch (plural); map (plural); qring (plural); ring (plural).

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