Every matrix in Macaulay must have a base ring. This base ring is a polynomial ring in a certain number of variables, over a prime field of characteristic p ;SPMgt; 0, for some p. It is possible to work modulo an ideal (see the qring command in section 5.3).
There may be many rings defined at any instant, but one of these rings is defined as the ``current ring''. For commands which input polynomials (e.g. poly, ideal, mat, sparse), this current ring is used as the base ring. Most commands set the current ring, such as add (add 2 matrices). You may set it explicitly using the setring command.
If a matrix is used as an argument to a Macaulay command which expects a ring, the base ring of the matrix is used. These commands include pring, setring, ring-sum, and the ring map commands (Chapter 9).
When a ring is created, say with the name ``r'', the returned value is actually a 1 by n matrix consisting of all the variables in the new ring.