Multiplicative homomorphisms of matrices

Abstract

G will denote a system closed under a multiplication. An element eEzG is called an identity if ae=ea=a for every aCG. An element Oz-G is called a null element if Oa=aO=O for every aEzG. Clearly e and 0 are unique if they exist; e=O if and only if G has just one element. A square root of the identity is an element qCG such that q2=e. Let HCG be the set consisting of the square roots of the identity in G and the null element if it exists. We assume throughout that the elements of H commute with each other. If G is a ring with identity and without divisors of zero and with ring multiplication as multiplication in G, then H consists of 0, e, e and these commute with every element of G, for if q2=e, (q-e)(q+e)=O and q= ?e. R will always denote a ring with identity, and 9& will denote the set of n Xn matrices with elements in R. Let Mi(c), Ei1, A ii(c) (i6j) be the matrices resulting respectively from the identity matrix I by multiplying row i by c, interchanging rows i and j, and adding row i multiplied by c to row j; these will be called elementary matrices. Let 9n* denote the set of matrices in 9Yn which are products of elementary matrices. For some rings R, On* = 9)n; if R is such a ring and 0 is a homomorphism of R onto a ring R', then 9N' * = 9J1' where the prime refers to matrices with elements in R'. For 0 induces in a natural way a homomorphism 0 of On onto ' (merely let 0 act on each element of the matrix) in which the image of an elementary matrix is elementary. Suppose that a nonnegative integral absolute value I a is defined in R subject only to the conditions that for every bXO and a in R, a=bq+r and a=q'b+r' where I|r, |r'| <Ibi. Then the usual procedure can be used to reduce a matrix in En to diagonal form by left and right multiplications by elementary matrices with inverses; see [1, vol. 2, p. 120 ff.]. A diagonal matrix is a product of elementary matrices Mi(c) and the inverse of an elementary matrix is elementary if it exists, hence if R has an absolute value as above, On* = 9n. A skew field or field or any euclidean ring admits such an absolute value. If a ring R has such an absolute value and ,B is a homomorphism of R onto a ring S, then for sES define IsI =min Inr for ,B(r)=s; this gives S an absolute value with the above properties. A mapping 1 of 91ThI or 9 * into G such that N(BC) =4 (B) (C) for every B, CGOn or 9n* respectively, will be called a multiplica-