fm S lF[a;o,a;i,... ,x„] is a projective variety of P G ( n , F ) , of which the affine portion is W nU = V. Mathematical Background 31 The above definitions of affine and projective varieties arc given in terms of a finite set of polynomials. 76 sliows that varieties are in fact defined by polynomial ideals. 76 Let / be an ideal of F [ a ; i , . . ,a;„]. If V(/) denotes the set { ( a i , .

We can also classify mappings between two algebras in the usual way, so an algebra homomorphism is a mapping that is both a ring homomorphism and a vector space homomorphism. 55 The ring of polynomials F [ a ; i , . . 40). Thus F [ x i , . . ,a:„] forms an F-algebra, known as a polynomial algebra. 47). Matrix multiplication is an associative bilinear mapping on yW„(F). Thus A^„(F) forms an F-algebra of dimension n^. The set 'Dn{¥) of n x n diagonal matrices over F forms a subalgebra of Mn{¥) of dimension n.

54 Suppose ^ is a vector space over a field F with a multiplication operation ^ x ^ —> ^ . If this multiplication operation is associative and a bilinear mapping on the vector space A, then A is an (associative) ¥-algebra, or more simply an algebra. Informally, we can regard an algebra as a vector space that is also a ring. The dimension of the algebra A is the dimension of yl as a vector space. T h e subset ,4' C ^ is a subalgebra of A if A! is an algebra in its own right, and A' is an ideal subalgebra if it is also an ideal of the ring A.