Each user defined variable has a type. The possible types of variables are: integer, ring, matrix (defined over a given ring), standard basis, and computation.
Rings are a basic concept of Macaulay. The major type of data which Macaulay manipulates are matrices with polynomial entries. In order to optimize calculations every matrix must be defined over a base ring. A ring contains necessary information such as the number of variables, their names, and the monomial order being used. The system uses this information to compact the monomials into as small a space as possible. See Chapter 5 for details.
In Macaulay, an ideal is simply a matrix with exactly one row. This means that even if the entries of two different matrices M, N with one row generate the same ideal, M and N are considered different.
A ring map is also represented as a matrix with one row (see Chapter 9).
Computations include standard (Gröbner) bases, syzygies, and finite free resolutions (see Chapter 7). A computation is made up of one or more standard bases. A standard basis is a matrix, together with a standard basis of the submodule generated by the columns of the matrix (see Chapter 6).
Variables can be created using the following commands: int creates an integer, ring creates a ring. ideal, mat, sparse, and poly create matrices, rmap, imap create ring maps, and std, nres, res and the rest of the commands of Chapter 7 create computations and standard bases.