Overview

Rings of various kinds form the richest source of algebraic structures in Magma. Tables 1 and 2 list the most important types.

Symbol         Description               Category
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Z              ring of integers          RngInt
Z/mZ           ring of residue classes   RngRes
R[x]           univariate poly. ring     RngUPol
F[x]/f(x)      univ. poly. factor ring   RngUPolRes
R[x_1,...,x_m] multivariate poly. ring   RngMPol
R[[x]]         power series ring         RngSer
O              order in a number field   RngOrd
\Z_p           p-adic ring               RngPad
R_m            local ring                RngLoc
V              valuation ring            RngVal
--------------------------------------------------
Q              rational field            FldRat
F_q            finite field              FldFin
F(x_1,...,x_m) rational function field   FldFun
F((x))         field of Laurent series   FldPow
Q(sqrt(D))     quadratic number field    FldQuad
Q(zeta_n)      cyclotomic number field   FldCyc
Q(alpha)       number field              FldNum
Q_p            p-adic field              FldPad
Q_p(alpha)     local field               FldLoc
R              real field                FldRe
C              complex field             FldCom
--------------------------------------------------

The list of rings in Table 1 is not exhaustive, for two reasons. In the first place, some rings have been categorized differently, because their module structure or algebra structure seems pre-eminent; thus matrix rings and finitely presented algebras appear (more or less arbitrarily) in the Part on Algebras, and vector spaces and their generalizations appear in the Module Part. (Also, rings of class functions appear in the Part on Groups.) Furthermore, certain general constructions (such as sub) allow the user to define rings that do not appear in the above list, most notably subrings of \({\mathbb{Z}}\). Looking at the table it may seem that all rings in Magma are commutative and unital. This is not the case (even though it would have made life much easier); since polynomial rings and the like can be defined over any coefficient ring, the matrix rings and finitely presented algebras not listed here that are not generally commutative, allow the construction of non-commutative rings. Furthermore, the sub constructor allows the creation of rings without 1; certain functions for the construction of new rings from old ones do not allow such non-unital coefficient rings. In this Chapter we give an overview of the various types and the relations between them. Moreover, we describe the important principles underlying the rules for coercion of elements of one ring into another. This Chapter also describes the common functions for all types of rings (and their elements), and subsequent Chapters deal with particular categories of rings, as indicated by the table.