Rings of various kinds form the richest source of algebraic structures in Magma. The tables list the most important types.
Symbol Description Category -------------------------------------------------- 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 the table 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 are classified in the online help (more or less arbitrarily) as algebras, and vector spaces and their generalizations as modules. (Also, rings of class functions are described under the node 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 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.