ring3 http://www.openmath.org/cd/ring3.ocd 2006-06-01 2004-06-01 1 1 experimental A CD of functions for basic constructions in ring theory. The quaternion definition is still very shaky. Written by Arjeh M. Cohen 2004-02-25 is_ideal The binary boolean function whose value is true if and only if the second argument is an ideal of the second. If is_ideal(S,I) then I is a nonempty set of elements of S and I is a subgroup of the additive group of S and closed under multiplication by elements of S. ideal This symbol represents a binary function. The first argument is a ring R and the second argument is a list or a set. When evaluated on R and such a second argument, the function represents the ideal in R generated by the entries of the list or set. The ideal in the free ring on the letters a, b generated by a*b-b*a: kernel This symbol represents a unary function. Its argument is a ring homomorphism f : R -> S. When evaluated on f, the function represents the kernel in R of f, that is, the subset {x in R | f(x) = 0}. The kernel of a ring homomorphism is an ideal. principal_ideal This symbol represents a binary function. The first argument is a ring R and the second argument is an element of R. When evaluated on R and such a second argument, the function represents the ideal in R generated by the second argument. The ideal in the free ring over the rationals on the letters a, b generated by a*b-b*a: free_ring This symbol represents a binary function. The first argument should be a ring and the second a list or a set. When evaluated on such arguments R and L, the function represents the free ring over R generated by the elements (or entries) of L. This ring can also be viewed as the ring of non-commutative polynomials over R with variables the elements of L. The free ring over R on the letters a, b: poly_ring This symbol represents a binary function. The first argument should be a ring and the second a variable. When evaluated on such arguments R and X, the function represents the free commutative ring over R generated by X. This ring can also be viewed as the ring of polynomials over R with indeterminate X. The polynomial ring over R with indeterminate X: m_poly_ring This symbol represents a binary function. The first argument should be a ring and the second a list or a set. When evaluated on such arguments R and L, the function represents the free commutative ring over R generated by the elements (or entries) of L. This ring can also be viewed as the ring of polynomials over R with variables the elements of L. The polynomial ring over R with variables a, b: matrix_ring This symbol represents a binary function. The first argument is a positive integer n, the second is a ring R. When evaluated on such argument n and R, the function represents the ring of n x n matrices over R. The ring of 1 x 1 matrices over R is isomorphic to R. 1 direct_product This is a symbol with two or more arguments, all of which are rings. It denotes the ring that is the direct product of its arguments. direct_power This is a symbol with two arguments. The first argument should be a ring S and the second argument a positive integer n. It denotes the direct product of n copies of S. 0 0 0 2 quotient_ring This is a binary function, whose first argument is a ring R and whose second argument is an ideal I of R. When applied to R and I, it denotes the quotient ring of R by I. The carrier of the ring of integers modulo 2 is introduced as Zm(2) in the CD setname2. The ring can also be defined as follows. 0 1 0 1 2 The ring (Z/2Z)[x]/(x^2+x+1) 2 2 2 1 Using the xref mechanism it can also be represented as 2 2 1 multiplicative_group This is a unary function, whose argument is a ring R. When applied to R, it denotes the group of invertible elements of R with respect to the multiplication on R. The multiplicative group of the ring R is the group of invertible elements of the multiplicative monoid of R. invertibles This is a unary function, whose argument is a ring R. When applied to R, it denotes the set of invertible elements of R with respect to the multiplication on R. The carrier of the multiplicative group of the ring R is the set of invertible elements of R. integers This is a symbol representing the ring of integers. The ring of integers is (Z, +,0,-,*,1), where +,-,* are the standard arithmetic operations. 0 1