ring2 http://www.openmath.org/cd/ring2.ocd 2006-06-01 2004-06-01 1 1 experimental Basic functions for homomorphisms in ring theory Initiated by Arjeh M. Cohen 2004-02-25 is_homomorphism This symbol is a boolean function with three arguments. The first and arguments are rings M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a ring homomorphism from M to N. If is_homomorphism(M,N,f) then, for each pair of elements x, y of M, we have f(x * y) = f(x) * f(y). is_isomorphism This symbol is a boolean function with three arguments. The first and arguments are rings M, N, the third is a map f from the element set of M to the element set of N. When applied to M, N, and f, it denotes that f is a ring isomorphism from M to N. This means that f is a homomorphism from M to N, that f is bijective, and that its inverse is a homomorphism from N to M. is_endomorphism This symbol is a boolean function with two arguments. The first argument is a ring M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes that f is a ring endomorphism from M to M. If is_endomorphism(M,f) then is_homomorphism(M,M,f) is_automorphism This symbol is a boolean function with two arguments. The first is a ring M, the second is a map f from the element set of M to the element set of M. When applied to M and f, it denotes a ring automorphism f of M. If is_automorphism(M,f) then is_isomorphism(M,M,f) left_multiplication This symbol is a function with two arguments, which should be a ring M and an element x of M. When applied to M and x, it denotes left multiplication on M by x. left_multiplication(M,x) (y) = x * y. right_multiplication This symbol is a function with two arguments, which should be a ring M and an element x of M. When applied to M and x, it denotes right multiplication on M by x. right_multiplication(M,x) (y) = y * x. isomorphic This symbol is a Boolean function with n arguments, n at least 2, which are rings. When applied to M_1, ..., M_n, it denotes the fact that there is an isomorphism from each M_i to each M_j.