ring4 http://www.openmath.org/cd/ring4.ocd 2006-06-01 2004-06-01 1 1 experimental A CD of functions for further basic properties of rings Written by Arjeh M. Cohen 2004-02-25 is_maximal_ideal The binary boolean function whose value is true iff the second argument is a maximal ideal of the first. is_prime_ideal The binary boolean function whose value is true iff the second argument is a prime ideal of the first. is_domain This symbol represents a boolean unary function. The argument is a ring R. When evaluated on R, the function returns true if R is a domain and false otherwise. A domain is a commutative ring without zero divisors. is_field This is unary boolean function whose argument should be a ring R. The value is true if and only if the ring is commutative and every nonzero element has a multiplicative inverse. If is_commutative(G) and for all a in carrier(G) there is b in carrier(G) such that a*b = identity(G), then is_field(G). is_zero_divisor This symbol represents a boolean binary function. The first argument is a ring R, the second is an element x of R. When evaluated on R and x, the function returns true if x a zero divisor and nonzero in R. An element x of a ring R is a zero divisor if and only if it nonzero and there is a nonzero y in R such that x * y = 0 or y * x = 0.