Neal H. McCoy

Annihilators in Polynomial Rings

It is known that if f is a divisor of zero in the polynomial ring R [x], where R is a commutative ring, there exists a non-zero element c of R such that cf=0. Proofs of this theorem have been given by McCoy [3], Forsythe [2], Cohen [1], and Scott [4]. It is clear that the theorem as stated does not immediately generalize to polynomials in more than one indeterminate. Moreover, it has been pointed out in Problem 4419 of this MONTHLY (1950, p. 692 and 1952, p. 336) that the theorem itself is not necessarily true for noncommutative rings. The purpose of this note is to obtain a suitable generalization of this result to the case of an arbitrary polynomial ring in any finite number of indeterminates. Let R be an arbitrary ring and R [x1, . . ., X] the ring of polynomials in the indeterminates xi, * * *, Xk, with coefficients in R. If A is a right ideal in R[x1, * .. ., xk], let us denote by Ar the set of right annihilators of A, that is, the ideal consisting of all elements h of the ring R [x1, * .. , Xk] such that Ah = 0. We shall prove the following theorem.

On quasi-commutative matrices

where c is a scalar matrix. It is well knownf that a relation of this type can not be satisfied by finite matrices. However, in the calculation of commutation formulas for polynomials in p and q no use is made of the fact that c is a scalar but merely that it is commutative with both p and q.% And there do exist pairs of finite matrices x, y of the same order such that xy—yx is not zero and is commutative with both x and y. Such matrices will be called quasi-commutative matrices and either may be said to be quasi-commutative with the other. In a certain sense the algebra of polynomials in a pair of quasi-commutative matrices is homeomorphic to the algebra arising in quantum mechanics. It is hoped to discuss such algebras in some detail in a later paper. In the present paper we shall make a brief study of quasi-commutative matrices whose elements belong to the complex number field. The concept of quasi-commutativity is an extension or generalization of commutativity, and as would be expected, some of the results obtained are generalizations of known theorems concerning commutative matrices. The problem of determining quasi-commutative matrices is that of finding matrices x, y, z (^0) which satisfy the equations

On the resultant of a system of forms homogeneous in each of several sets of variables

be a set of n general forms homogeneous in the n variables xi, x2, *, Xn; to determine the polynomial in the coefficients of these forms whose vanishing is a necessary and sufficient condition that the forms (1) simultaneously vanish for a set of values, not all zero, of the variables xi, x2, * , x,. This polynomial is called the resultant of the system of forms (1). From this standpoint a numerical factor in the resultant is of no consequence though in certain cases it is desirable to introduce some convention as to such a factor. The most important properties of the resultant of the system (1) are well known and have been obtained by various authors in a variety of ways.t We give a brief account of the method used by Konigt as it is of particular importance in the sequel. Let us denote by

A generalization of Ostrowski’s theorem on matric identities

In this theorem it is tacitly assumed that the elements of the matrices as well as the coefficients of the polynomials are real or complex numbers. In Theorem 3 below we find an extension of the first part of Theorem 1, valid if the elements and coefficients are in an arbitrary commutative ring R with unit element 1. To generalize the second part of the theorem, we find it necessary to make an additional restriction on Ry namely, that there exists no nonzero polynomial </>(X)