factorcharacteristic

Factorcharacteristic is a term used in algebra to denote the characteristic of a factor ring, that is, the quotient of a ring by one of its ideals. More precisely, for a ring R with identity and an ideal I ⊆ R, the factorcharacteristic of the pair (R, I) is the characteristic of the quotient ring R/I, denoted char(R/I). While the concept is straightforward, many texts refer to it simply as the characteristic of the quotient ring.

Computation and interpretation: The characteristic of R/I is the smallest positive integer n such that n·1_R lies in I; if no such n exists, char(R/I) = 0, indicating infinite characteristic. In particular, if char(R) is finite n, then char(R/I) divides n. If char(R) = 0, char(R/I) may be finite or infinite, depending on I.

Examples: For R = Z and I = nZ, R/I ≅ Z/nZ, so char(R/I) = n. For R = F_p[x] and I = (f) where f is any nonzero polynomial, the quotient R/I has characteristic p, the same as the base ring F_p.

Properties and usage: The factorcharacteristic interacts with quotient operations, helping classify how changing ideals affects arithmetic in the quotient. It is useful in areas such as algebraic geometry and number theory when analyzing reductions modulo primes or modulo ideals, and in studying the behavior of rings under quotient constructions.

Note: The label factorcharacteristic is not universally standard in literature; the underlying concept is the characteristic of a quotient ring.