Simple ring

Abstract: For each matrix, whose elements belong to a commutative ring with an identity element, there is defined a free complex. This complex is a generalization of the standard Koszul complex, which corresponds to the case of a matrix with only a single row. The applications are to certain ideals defined by the maximal subdeterminants of a matrix. It is found that such an ideal has finite projective dimension whenever its grade reaches a certain greatest value (depending on the dimensions of the matrix) and that, in these circum stances, the complex provides a free resolution of the correct length. For semi-regular ( = M acaulayCohen) rings this leads to a theorem on unmixed ideals. In the case of arbitrary Noetherian rings, a general theorem on rank is proved.

Abstract: We give an algorithm which represents the radical J of a finitely generated differential ideal as an intersection of radical differential ideals. The computed representation provides an algorithm for testing membership in J. This algorithm works over either an ordinary or a partial differential polynomial ring of characteristic zero. It has been programmed. We also give a method to obtain a obtain a characteristic set of J, if the ideal is prime.

Abstract: Introduction. The purpose of this paper is to provide a fresh outlook to various questions on rings with polynomial identity by examining the centers of such rings. This approach yields the interesting result that any nonzero ideal of a semiprime ring with polynomial identity intersects the center nontrivially (Theorem 2). There are at least two interesting consequences to Theorem 2: a generalization of Wedderburn's theorem (any semiprimitive ring with polynomial identity, whose center is a field, is simple) and a strengthening of Posner's theorem [1] (any prime ring with a polynomial identity has a simple ring of quotients whose center is the quotient field of the center of the prime ring). The proofs are elementary modulo Jacobson [3]. Of course rings are not necessarily commutative and for the sake of simplicity we assume a unit 1. The key argument in this paper is an application of Formanek's central polynomials for matrix algebras over a field, whose important properties are [2] : Let Mn be an n x n matrix algebra over an arbitrary field. Then there exists a polynomial gn(Xl9.. .,Xm) which has coefficients in Z; is homogeneous (degree > 0) in every variable and linear in all but the first variable; takes values in the center for every specialization in Mn; and is nonvanishing for some specialization.

Abstract: An abstract ring in which all finitely generated ideals are principal will be called an F-ring. Let C(X) denote the ring of all continuous real-valued functions defined on a completely regular (Hausdorff) space X. This paper is devoted to an investigation of those spaces X for which C(X) is an F-ring. In any such study, one of the problems that arises naturally is to determine the algebraic properties and implications that result from the fact that the given ring is a ring of functions. Investigation of this problem leads directly to two others: to determine how specified algebraic conditions on the ring are reflected in topological properties of the space, and, conversely, how specified topological conditions on the space are reflected in algebraic properties of the ring. Our study is motivated in part by some purely algebraic questions concerning an arbitrary F-ring S-in particular, by some problems involving matrices over S. Continual application will be made of the results obtained in the preceding paper [4]. This paper will be referred to throughout the sequel as GH. We wish to thank the referee for the extreme care with which he read both this and the preceding paper, and for making a number of valuable suggestions. The outline of our present paper is as follows. In ?1, we collect some preliminary definitions and results. ?2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring). The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, a-compact space (e.g., the reals), then fX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are: (i) for every f C(X), there exists k E C(X) such that f k Jf J; (ii) for every maximal ideal M of C(X), the intersection of all the prime ideals of C(X) contained in M is a prime ideal. In ??3 and 4, we study Hermite rings and elementary divisor rings(2).